Lagrangian Reduction by Stages - Caltech Computingmarsden/volume/LRBS/lrbs.pdfPage v Preface This booklet studies the geometry of the reduction of Lagrangian systems with symmetry

Page i

Lagrangian Reduction by Stages

Memoirs of the American Mathematical Society

152, no. 722, July 2001, 108 pp.

(Recieved by the AMS, April, 1999; Updated January 22, 2009).

Hernan CendraDepartamento de Matematica

Universidad Nacional del Sur, Av. Alem 12548000 Bahia Blanca, [email protected] 1

Jerrold E. MarsdenControl and Dynamical Systems 107-81

California Institute of Technology Pasadena, CA [email protected] 2

Tudor S. RatiuDepartement de Mathematiques

Ecole Polytechnique Federale de Lausanne CH 1015 Lausanne, [email protected] 3

1 October 2009

1The research of HC was partly done during a sabbatical stay in Control and DynamicalSystems at the California Institute of Technology.

2The research of JEM was supported in part by the California Institute of Technologyand NSF Grants DMS-9802106 and ATM-9873133.

3The research of TSR was supported in part by NSF Grant DMS-9802378 and FNSGrant 21-54138.98.

mailto:[email protected]



ii

Page iii

Contents

Preface v

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The Main Results of This Paper. . . . . . . . . . . . . . . . . . . . 31.3 Future Work and Related Issues. . . . . . . . . . . . . . . . . . . . 4

2 Preliminary Constructions 72.1 Notation and general assumptions. . . . . . . . . . . . . . . . . . . 72.2 Connections on Principal Bundles. . . . . . . . . . . . . . . . . . . 92.3 Associated Bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 The Bundles TQ/G and T (Q/G)⊕ g . . . . . . . . . . . . . . . . . 16

3 The Lagrange–Poincare Equations 193.1 The Geometry of Variations . . . . . . . . . . . . . . . . . . . . . . 193.2 The Euler–Lagrange and Euler–Poincare Operators . . . . . . . . . 253.3 The Lagrange–Poincare Operator . . . . . . . . . . . . . . . . . . . 333.4 The Reduced Variational Principle . . . . . . . . . . . . . . . . . . 39

4 Wong’s Equations and Coordinate Formulas 434.1 Wong’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 The Local Vertical and Horizontal Equations . . . . . . . . . . . . 45

5 The Lie Algebra Structure on Sections of the Reduced Bundle 515.1 The Bundle T (Q/G)⊕ g Revisited . . . . . . . . . . . . . . . . . . 51

iv Contents

5.2 The Lie Algebra of Sections of T (Q/G)⊕ g . . . . . . . . . . . . . 54

6 Reduced Tangent Bundles 616.1 The Geometry of Lagrange–Poincare Bundles . . . . . . . . . . . . 626.2 Reduction of Lagrange–Poincare Bundles . . . . . . . . . . . . . . . 736.3 Reduction by Stages of objects of LP . . . . . . . . . . . . . . . . . 846.4 The Subcategory RT and Reduction by Stages . . . . . . . . . . . 92

7 Further Examples 1017.1 Semidirect Products . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.2 Central Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.3 Rigid Body with Rotors . . . . . . . . . . . . . . . . . . . . . . . . 1047.4 Systems Depending on a Parameter . . . . . . . . . . . . . . . . . . 105

8 The Category LP∗ and Poisson Geometry 1118.1 The Poisson Bracket on Duals of Objects of LP . . . . . . . . . . . 1118.2 Poisson Reduction in the Category LP∗ . . . . . . . . . . . . . . . 117

Page v

Preface

This booklet studies the geometry of the reduction of Lagrangian systems withsymmetry in a way that allows the reduction process to be repeated; that is, itdevelops a context for Lagrangian reduction by stages. The Lagrangian reductionprocedure focuses on the geometry of variational structures and how to reducethem to quotient spaces under group actions. This philosophy is well known for theclassical cases, such as Routh reduction for systems with cyclic variables (wherethe symmetry group is Abelian) and Euler–Poincare reduction (for the case inwhich the configuration space is a Lie group) as well as Euler-Poincare reductionfor semidirect products.

The context established for this theory is a Lagrangian analogue of the bundlepicture on the Hamiltonian side. In this picture, we develop a category that includes,as a special case, the realization of the quotient of a tangent bundle as the Whitneysum of the tangent of the quotient bundle with the associated adjoint bundle.The elements of this new category, called the Lagrange–Poincare category, haveenough geometric structure so that the category is stable under the procedure ofLagrangian reduction. Thus, reduction may be repeated, giving the desired contextfor reduction by stages. Our category may be viewed as a Lagrangian analog of thecategory of Poisson manifolds in Hamiltonian theory.

We also give an intrinsic and geometric way of writing the reduced equations,called the Lagrange–Poincare equations, using covariant derivatives and connec-tions. In addition, the context includes the interpretation of cocycles as curvaturesof connections and is general enough to encompass interesting situations involvingboth semidirect products and central extensions. Examples are given to illustratethe general theory.

vi Preface

In classical Routh reduction one usually sets the conserved quantities conjugateto the cyclic variables equal to a constant. In our development, we do not requirethe imposition of this constraint. For the general theory along these lines, we referto the complementary work of Marsden, Ratiu and Scheurle (2000), which studiesthe Lagrange–Routh equations.

Received by the editor April 12, 1999 and in revised form March 15, 2000.1991 Mathematics Subject Classification. Primary 37J15; Secondary 70H33, 53D20.

Key words and phrases. Lagrangian reduction, mechanical systems, variational principles,

symmetry.

Page 1

1Introduction

Reduction theory for mechanical systems with symmetry has its origins in theclassical works of Euler, Lagrange, Hamilton, Routh, Jacobi, Liouville and Poincare,who studied the extent to which one can reduce the dimension of the phase space ofthe system by making use of any available symmetries and associated conservationlaws. Corresponding to the main two views of mechanics, namely Hamiltonian andLagrangian mechanics, one can also adopt two views of reduction theory.

In symplectic and Poisson reduction, which are now well developed and muchstudied subjects, one focuses on how to pass the symplectic two form and thePoisson bracket structure as well as any associated Hamiltonian dynamics to aquotient space for the action of a symmetry group (see, for example, Meyer (1973),Marsden and Weinstein (1974), Marsden and Ratiu (1986) and the expositions inAbraham and Marsden (1978), Arnold (1989), Libermann and Marle (1987) andMarsden (1992)).

In Lagrangian reduction theory, which proceeds in a logically independent way,one emphasizes how the variational structure passes to a quotient space (see, forexample, Cendra and Marsden (1987), Cendra, Ibort and Marsden (1987), Mars-den and Scheurle [1993a, 1993b], Bloch, Krishnaprasad, Marsden and Ratiu (1996),Cendra, Holm, Marsden and Ratiu (1998), Holm, Marsden and Ratiu (1998a), Jal-napurkar and Marsden (2000) and Marsden, Ratiu and Scheurle (2000)). Of course,the two methodologies are related by the Legendre transform, although not alwaysin a straightforward way.

The main purpose of this work is to further the development of Lagrangian reduc-tion theory. There are several aspects to this program. First, we provide a contextthat allows for repeated Lagrangian reduction by the action of a symmetry group.Second, we provide the geometry that is useful for the expression of the reduced

2 1. Introduction

equations, called the Lagrange–Poincare equations. Further details concerning themain results of this work are given shortly.

1.1 Background

As we have mentioned, in the last few years there has been considerable activity inthe area of Lagrangian reduction in which one focuses on the reduction of variationalprinciples. We shall review the background for this theory briefly, starting with thebest known classical results.

Classical Cases.

Several classical instances of Lagrangian reduction are well known, such as Routhreduction which was developed by Routh (1877) in connection with his studies ofthe stability of relative equilibria. Routh began the development of what we wouldcall today Lagrangian reduction for Abelian groups. One thinks of this case astreating mechanical systems with cyclic variables.

Another fundamental case is that of Euler–Poincare reduction, which occurs forthe case in which the configuration space is a Lie group. One thinks of this caseas primarily intended for systems governed by Euler equations, such as those of arigid body and a fluid. This case has its origins in the work of Lagrange (1788) andPoincare (1901a). Both of them clearly had some idea of the reduction process.

In these classical works, many important ideas were developed. However, forboth of these cases, some of the clarifications and generalizations are remarkablyrecent. For example, only in Bretherton (1970) was the reduced variational principleestablished for the fluid equations, but this was done by ad hoc rather than generalmethods. Both the intrinsic (coordinate free) formulation of Routh reduction aswell as the general formulation of Euler–Poincare reduction in terms of variationalprinciples were given in Marsden and Scheurle [1993a, 1993b]). The Euler–Poincarecase was further developed in Bloch, Krishnaprasad, Marsden and Ratiu (1996). Anexposition of Lagrangian reduction for both the Routh and Euler–Poincare casescan be found in Marsden and Ratiu (1999).

Semidirect Product Theory.

Another well developed subject in Hamiltonian reduction theory is that of semidi-rect product theory. This theory has its origins in the work of Guillemin and Stern-berg (1980) (see also Guillemin and Sternberg (1984)), Ratiu [1980a; 1981; 1982a],and Marsden, Ratiu, and Weinstein [1984a, 1984b]. This theory has many interest-ing applications, such as to the heavy top, MHD, and the dynamics of underwatervehicles (Leonard and Marsden (1997)). This semidirect product theory was a directprecursor to the development of symplectic reduction by stages (Marsden, Misiolek,Perlmutter and Ratiu [1998, 2000]).

1.1 Background 3

Lagrangian analogues of the semidirect product theory were developed in Holm,Marsden and Ratiu (1998a) with applications to many fluid mechanical problemsof interest. The point of view was to extend the Euler–Poincare theory to thecase of systems such as the heavy top and compressible flows in which there areadvected parameters or fields. This methodology was applied to the case of theMaxwell–Vlasov equations by Cendra, Holm, Hoyle and Marsden (1998); Cendra,Holm, Marsden and Ratiu (1998) showed how it fits into the general framework ofLagrangian reduction.

Symplectic versus Poisson Reduction.

We should emphasize that in the framework of the theory of Poisson manifolds,Poisson reduction (in the naive sense of just taking nonsingular quotients, not themore sophisticated sense of Marsden and Ratiu (1986)) by stages is quite simple,while in the framework of symplectic manifolds, symplectic reduction by stagesis more sophisticated. On the other hand, if one wants to study the reduction ofcotangent bundles and to retain as much of this structure as possible, then eventhe Poisson point of view is quite nontrivial.

The Lagrangian analogue of symplectic reduction is nonabelian Routh reduction(Marsden and Scheurle [1993a, 1993b], Jalnapurkar and Marsden (2000) and Mars-den, Ratiu and Scheurle (2000)) and its full development in the context of reductionby stages is the subject of a future publication. The present work represents a La-grangian analogue of the Poisson version of reduction by stages but keeping thestructure of the tangent bundle as much as possible. One of the things that makesthe Lagrangian side interesting has been the lack of a general category that isthe Lagrangian analogue of Poisson manifolds. Such a category, that of Lagrange–Poincare bundles is given in §8, with the tangent bundle of a configuration manifoldand a Lie algebra as its two most basic examples. We also develop the Lagrangiananalogue of reduction for central extensions and, as in the case of symplectic re-duction by stages, cocycles and curvatures enter in this context in a natural way.

Nonholonomic Mechanics.

The ideas of geometric mechanics and Lagrangian reduction have had a signifi-cant impact on the theory of nonholonomic systems (such as mechanical systemswith rolling constraints), as in Bloch, Krishnaprasad, Marsden and Murray (1996)and Koon and Marsden [1997b, c, 1998], Marsden, Ratiu, and Weinstein [1998,1994] Bloch and Crouch (1999) and Lewis [1996, 2000]. These references also de-velop Lagrangian reduction methods in the context of nonholonomic mechanicswith symmetry. These methods have been quite useful in many control problemsand in robotics. The techniques of the present paper can be used to give an in-trinsic geometric meaning to the reduction of the Lagrange-d’Alembert equationsof nonholonomic mechanics. This is the subject of the work Cendra, Marsden andRatiu (2000).

4 1. Introduction

Control Theory.

Geometric mechanics and Lagrangian reduction theory has also had a significantimpact on control theory, including stabilization (as in Bloch, Leonard and Marsden(2000) and Jalnapurkar and Marsden (1999) as well as on optimal control theory;see Vershik and Gershkovich [1988, 1994], Bloch and Crouch [1993, 1994, 1995]Montgomery [1990, 1993], Koon and Marsden (1997a) and references therein.

1.2 The Main Results of This Paper.

We now give a few more details concerning the main results. The first of these,given in §5, develops the theory of Lagrange-Poincare bundles, which enable oneto perform Lagrangian reduction in stages. Lagrange-Poincare bundles may be re-garded as the Lagrangian analogue of a Poisson manifold in symplectic geometry.Lagrange–Poincare bundles include, of course, the case of reduced tangent bundles(TQ)/G in which we take the quotient of the tangent bundle of the configurationspace Q by the action of a Lie group G on Q. This in turn, includes importantexamples such as Euler–Poincare reduction for the special case Q = G, a Lie group,in which case, (TQ)/G = g, the Lie algebra of G. Euler–Poincare reduction is nowa textbook topic that can be found in Marsden and Ratiu (1999). We show thatwhen a general tangent bundle is reduced by a group action, one ends up in thecategory of Lagrange-Poincare bundles.

We mention that Lagrange-Poincare bundles are in particular, Lie algebroids butcarry additional structure. We will not use any theory of groupoids or algebroids inthis work, but we will comment on part of the literature in the body of the work.

The Lagrange–Poincare equations are expressed using connections and curvature.These equations are obtained using the idea of reducing variational principles. TheLagrange-Poincare category is stable under reduction and the structure carriedby Lagrange–Poincare bundles is exactly what is needed to write the Lagrange-Poincare equations in a covariant form.

In §5.3 and §5.4 we show that if the symmetry group has a normal subgroup(i.e., one has a group extension), then reducing by the whole group is shown to beisomorphic to what one gets by reducing in stages, first by the normal subgroupfollowed by reduction by the quotient group. This result is a Lagrangian analogueof doing Poisson reduction by stages, but keeping track of the local structure ofPoisson manifolds, as in the Lie–Weinstein theorem (see Weinstein (1983a)).

The theory we establish may be viewed as the Lagrangian analogue of the bundlepicture on the Hamiltonian side developed by Montgomery (1986) and Montgomery,Marsden and Ratiu (1984). This bundle picture was in turn influenced by work onWong’s equations for a particle in a Yang–Mills field, as studied by Sternberg (1977),Weinstein (1978), Montgomery (1984), and Koon and Marsden (1997a). As we shallsee in §3.3, this theory has a very beautiful Lagrangian analogue. In §6 we give anumber of additional examples. In future works we plan to establish additional linkswith the Hamiltonian side.

1.3 Future Work and Related Issues. 5

1.3 Future Work and Related Issues.

Our theory naturally suggests a number of additional things that warrant furtherinvestigation.

Geometric Phases.

The development of the theory of geometric phases in the Lagrangian context isnatural to develop given the relatively large amount of activity from the symplec-tic and Poisson point of view (see, eg, Marsden, Montgomery and Ratiu (1990),Marsden (1992), Blaom (2000) and references therein).

The paper of Marsden, Ratiu and Scheurle (2000) gives geometric phase formulasin the context of Routh reduction. The development of geometric phases by stageswould be of interest. In fact, the Lagrangian setting provides natural connectionsand also a natural setting for averaging which is one of the basic ingredients ingeometric phases.

Nonholonomic Mechanics.

As mentioned above, the work of Cendra, Marsden and Ratiu (2000) extends thenotion of Lagrange–Poincare bundles to those of Lagrange-d’Alembert-Poincarebundles that are appropriate for nonholonomic mechanics. This extension may beregarded as the Lagrangian analogue of the notion of an almost Poisson mani-fold (in which Jacobi’s identity can fail), as in Koon and Marsden (1998) andCannas da Silva and Weinstein (1999). Furthering the links with almost Poissonmanifolds and also developing a nonholonomic reduction by stages theory would ofcourse be of interest.

Further Relations with the Hamiltonian Side.

It would also be significant to investigate the precise relationship of the work herewith the Hamiltonian reduction by stages theory in more detail, in particular, therelation with symplectic reduction by stages applied to cotangent bundles. Thisrequires the nonabelian Routh reduction analogue of the work here, namely thatof Marsden, Ratiu and Scheurle (2000), which extends the work of Marsden andScheurle (1993a) and Jalnapurkar and Marsden (2000).

Relations with Poisson Geometry.

The Lie–Weinstein theorem states that a Poisson manifold is locally the productof a symplectic manifold and the dual of a Lie algebra. This paper develops a La-grangian category that locally looks like the dual of this local structure for Poissonmanifolds. Our bundles actually have more structure than this, which is impor-tant for carrying out the reduction, namely we also carry along a connection and a

6 1. Introduction

two-form that helps keep track of curvature (or magnetic terms). This structure isalso very useful for writing covariant versions of the reduced equations, that is, theLagrange–Poincare equations.

Variational Integrators and Discrete Reduction.

As Weinstein (1996) points out, there is a more general context for Lagrangianmechanics that also includes discrete mechanics in the sense of Veselov [1988, 1991]and Moser and Veselov (1991). Our category of Lagrange–Poincare bundles doesnot include this literally, but still, there is a well defined discrete analogue of thesebundles. This picture is useful in the understanding of the reduced Hamilton-Jacobiequation (see Ge and Marsden (1988)).

One of the interesting developments in symplectic integration algorithms has beenthe progress made in variational integrators. These are based on direct discretiza-tions of Hamilton’s principle following some of the ideas of Veselov (1988). See, forexample, Wendlandt and Marsden (1997), Marsden, Patrick and Shkoller (1998)and Kane et al. (2000). There is a very interesting discrete reduction theory forthese that is still under development. See Marsden, Pekarsky and Shkoller (1999),Bobenko and Suris [1999a, 1999b] and Jalnapurkar Jalnapurkar, Leok, Marsdenand West (2000).

Infinite Dimensional Examples.

In this paper we will be dealing with Lagrangian reduction theory in the contextof finite dimensional manifolds. Of course the theory formally applies to manyinteresting infinite dimensional examples. In the infinite dimensional context, manyof the expressions that appear here as pure partial derivatives must be written inthe notation of functional derivatives (see Marsden and Ratiu (1999) for some of thebasic examples, an explanation of the functional derivative notation, and additionalreferences to the literature).

Multisymplectic Context.

Another area of much current activity is that of multisymplectic geometry. See, forexample, Marsden and Shkoller (1999) and Marsden, Patrick and Shkoller (1998).This theory has both a Lagrangian and a Hamiltonian view and it has allowed,for example, a development of the Moser-Veselov theory to the context of PDE’s.Reduction theory in this context is in its infancy (see, for example, Marsden, Mont-gomery, Morrison and Thompson (1986) and Castrillon-Lopez, Ratiu and Shkoller(2000)). Obviously it would be of interest to develop such a theory from the La-grangian reduction point of view.

An exciting possible application where reduction by stages is involved is that ofvarious complex fluids, such as liquid crystals, where there is a group of particlerelabeling symmetries, as in fluids and plasmas, as well as an internal order param-

1.3 Future Work and Related Issues. 7

eter group. See Holm (2000). This sort of example would also provide an interestingcontext for multisymplectic reduction by stages!

8 1. Introduction

Page 9

2Preliminary Constructions

In this section we recall some results about bundles and connections that we will uselater. As a general reference, see Kobayashi and Nomizu (1963); however, the readershould be warned that various conventions and notations differ from this referenceand we shall point these out as we proceed. In general, we follow the conventionsof Abraham and Marsden (1978) and Abraham, Marsden and Ratiu (1988). Weprovide intrinsic proofs with a view to infinite dimensional generalizations andalso because of the insight they provide. Coordinate expressions are important inmany applications, and they can be readily provided (some of these may be foundin Marsden and Scheurle [1993a, 1993b] and Bloch, Krishnaprasad, Marsden andMurray (1996)).

2.1 Notation and general assumptions.

Maps. If f : A → B is a map between sets and C ⊂ A, we shall often writef : C → B instead of f |C : C → B, for short.

Manifolds.

Unless otherwise noted, for simplicity, manifolds are assumed to be C∞, as aremaps between them. While the manifolds will be assumed to be finite dimensional,many results can be easily generalized for infinite dimensions in a straightforwardmanner.

10 2. Preliminary Constructions

For a manifold Q, we let TQ be the tangent bundle of Q. An element of TqQ willbe denoted vq, uq, . . ., or (using a standard abuse of notation) by (q, q) or simplyby q.

Actions.

Unless otherwise noted, an action ρ : G×Q→ Q of a Lie group G on a manifold Q isassumed to satisfy the additional condition that, relative to this action, the manifoldQ becomes a principal bundle with structure group G, say πG(Q) : Q → Q/G. Inother words, we assume that Q/G is a manifold, that πG(Q) is a submersion andthat the action is free. As is well known (see, for example, Abraham and Marsden(1978) for the proof), if the action is both free and proper, then this hypothesis issatisfied.

We will often use the equivalent notations πG(Q)(q) = [q]G for the equivalenceclass of q ∈ Q. We will work with left actions in this paper unless explicitly notedotherwise; however, generalizations for right actions are straightforward.

The assumption of freeness of the action is, of course, a strong one and it corre-sponds, in the Hamiltonian case, to eliminating the case of singular reduction (seeSjamaar and Lerman (1991), Bates and Lerman (1997), Ortega (1998), and Ortegaand Ratiu [1997, 2001]). Similarly, on the Lagrangian side, this paper does not ad-dress the case of singular reduction. Of course, such questions are very interesting,but there is plenty to do even omitting that topic.

We will often use the equivalent notations ρ(g, q) ≡ ρg(q) ≡ ρq(g) ≡ g ·q ≡ gq forthe action of the group element g on the point q ∈ Q. However, the concatenationnotation gq will be used most commonly. The tangent lift of this action will bedenoted gvq, where vq ∈ TqQ. For any element ξ in the Lie algebra g of G, theinfinitesimal generator at q ∈ Q is denoted ξq ≡ ξQ(q) and it is defined, as usual,by

ξq :=d

dt

∣∣∣∣t=0

(exp tξ)q.

Bundles.

If π : P → Q is a fiber bundle and q ∈ Q, the fiber π−1(q) at q is sometimes denotedPq. If τi : Vi → Qi are vector bundles for i = 1, 2 and f : V1 → V2 is a vector bundlemap, the induced map on the zero-sections, which are usually identified with Qi,i = 1, 2, is denoted f0 : Q1 → Q2, or sometimes, by a slight abuse of notation,simply f .

Given two bundles πi : Pi → Q, i = 1, 2, the fiber product is the bundleπ1 ×Q π2 : P1 ×Q P2 → Q where P1 ×Q P2 is the set of all elements (p1, p2) ∈P1 × P2 such that π1(p1) = π2(p2) and the projection π1 ×Q π2 : P1 ×Q P2 → Qis naturally defined by π1 ×Q π2(p1, p2) = π1(p1) = π2(p2). The fiber is given by(π1 ×Q π2)−1(q) = π−1

1 (q)× π−12 (q).

A principal bundle is a manifold Q with a free left action G×Q→ Q of a Liegroup G, such that the natural projection π : Q→ Q/G is a submersion.

2.2 Connections on Principal Bundles. 11

The Whitney sum of two vector bundles τi : Vi → Q, i = 1, 2, over the samebase is their fiber product. It is a vector bundle over Q and is denoted V1⊕V2. Thisbundle is obtained by taking the fiberwise direct sum of the fibers of V1 and V2.

Sometimes, for purposes of uniformizing notation, we will consider a manifold Qas being identified with the vector bundle over Q whose fibers have dimension 0.Therefore if τ : V → Q is a vector bundle, Q⊕V is simply V , and an element q⊕ vof Q⊕ V satisfies τv = q.

Connections.

The word connection will be used in two different but standard senses in this paper.Sometimes it will mean a principal connection on a principal bundle and sometimeswill mean a connection on a vector bundle, usually denoted ∇ with the addition,sometimes, of some indexes to clarify the spaces on which the connection is defined.In any case, the context will always make it clear the sense in which we are using theword. We will recall some of the key notions and conventions used in the followingparagraphs.

2.2 Connections on Principal Bundles.

Horizontal and Vertical Spaces. Let π : Q → Q/G be a left principal bundle,where π is the canonical projection. Recall that a (principal) connection A onQ is a Lie algebra valued one form A : TQ→ g with the properties

(i) A(ξq) = ξ for all ξ ∈ g; that is, A takes infinitesimal generators of a given Liealgebra element to that element, and

(ii) A(Tqρg · v) = Adg(A(v)), where Adg denotes the adjoint action of G on g.

The restriction of a connection to the tangent space TqQ is denoted Aq. Recall thatconnections may be characterized by giving their vertical and horizontal spacesdefined at q ∈ Q by

Verq = KerTqπ, Horq = KerAq.

Thus, Ver(TQ) = ∪q∈Q Verq is the subbundle of vectors tangent to the group orbits.The vertical and horizontal components of a vector vq will be denoted Ver(vq)and Hor(vq) respectively. By definition,

Ver(vq) = A(vq)q and Hor(vq) = vq −A(vq)q.

This provides a decomposition TQ = Hor(TQ) ⊕ Ver(TQ) where Hor(TQ) =∪q∈Q Horq and Ver(TQ) are the horizontal and vertical subbundles of TQ, whichare invariant under the action of G. A vector is called horizontal if its verticalcomponent is zero; i.e., if A(vq) = 0 and it is called vertical if its horizontal com-ponent is zero; i.e., if Tqπ(vq) = 0. Note that Tqπ : Horq → Tπ(q)(Q/G) is anisomorphism.


Curvature.

The curvature of A will be denoted BA or simply B. By definition, it is the Liealgebra valued two form on Q defined by

B(uq, vq) = dA(Horq(uq),Horq(vq)),

where d denotes the exterior derivative.The reader should be aware that, although the notion of the curvature of a

given connection on a principal bundle is a well established and an essentiallyunique concept, there are many conventions related to various sign conventions anddefinitions of the wedge product and the exterior derivative. The one adopted herefor left actions is consistent with the one given in Abraham and Marsden (1978).

Cartan Structure Equations.

The Cartan structure equations state that for vector fields u, v (not necessarilyhorizontal) on Q, we have

B(u, v) = dA(u, v)− [A(u), A(v)], (2.2.1)

where the bracket on the right hand side is the Lie bracket in g. We write thisequation for short as

B = dA− [A,A].

Horizontal Lifts.

Given a vector X ∈ Tx(Q/G), and q ∈ π−1(x), the horizontal lift Xhq of X at q is

the unique horizontal vector in TqQ that projects via Tπ to the vector X(x); thatis, Xh

q ∈ (Tqπ)−1(X). We denote by Xh the vector field along π−1(x) formed byall horizontal lifts of X at points of π−1(x).

For any curve x(t) in Q/G, where t ∈ [a, b], the family of horizontal lifts is denotedxh. The definition is the following. For any point q0 ∈ π−1(x0), where x0 = x(t0),for some t0 ∈ [a, b], the horizontal lift of x(t), which at t = t0 coincides with q0, isuniquely determined by requiring its tangent to be a horizontal vector. This curveis denoted xhq0 and is defined on [a, b].

Consider a curve q(t), where t ∈ [a, b], and choose t0 ∈ [a, b]. Then there isa unique horizontal curve qh(t) such that qh(t0) = q(t0) and π (qh(t)) = π (q(t))for all t ∈ [a, b]. Therefore, we can define a curve gq(t), t ∈ [a, b] in G by thedecomposition

q(t) = gq(t)qh(t) (2.2.2)

for all t ∈ [a, b]. Evidently gq(t0) is the identity. Also, notice that if x(t) = π(q(t))and q0 = q(t0) then qh(t) = xhq0(t).

Lemma 2.2.1. For any curve q(t), t ∈ [a, b] in Q we have

A(q, q) = gqg−1q .

2.3 Associated Bundles. 13

Proof. We start with the equality gq(t)qh(t) = q(t). Differentiating this withrespect to t gives

gq(t)qh(t) + gq(t)qh(t) = q(t).

Here, the notation is interpreted in an obvious way; for example, for ug ∈ TgG andq ∈ Q, ugq means the derivative of the orbit map g 7→ gq in the direction of ug togive an element of TgqQ.

By definition of a horizontal vector, A (gq(t)qh(t)) = 0. Recall also that theconnection reproduces the Lie algebra element on infinitesimal generators (i.e.,A(ξq) = ξ for ξ ∈ g and q ∈ Q). In particular, for ξ = gqg

−1q and q(t) = gq(t)qh(t)

we getA(gq(t)qh(t)) = A(gq(t)g−1

q (t)gq(t)qh(t)) = gqg−1q ,

from which the result follows.

Curvature and Horizontal Lifts. For X1, X2 ∈ X∞(Q/G), let Xh1 , X

h2 ∈ X∞(Q)

be their horizontal lifts. Thus, Xhi and Xi are π-related, that is, Tπ Xh

i = Xi π,for i = 1, 2. However, the bracket operation of vector fields preserves π-relatedness(see, for example, Abraham, Marsden and Ratiu (1988)) and hence Tπ [Xh

1 , Xh2 ] =

[X1, X2]π. Thus, Hor[Xh1 , X

h2 ](q) and [X1, X2]h(q) are two horizontal vectors that

project by Tqπ to [X1, X2](π(q)) and hence they are equal. This proves the identity

Hor[Xh1 , X

h2 ] = [X1, X2]h. (2.2.3)

By Cartan’s structure equations, we get

B(q) (X1(q), X2(q)) q = (dA(q) (X1(q), X2(q))) q

= −(A([Xh

1 , Xh2

])(q))q

= Ver[Xh

1 , Xh2

](q)

= −[Xh

1 , Xh2

]+ Hor

[Xh

1 , Xh2

](q),

which shows, using (2.2.3), that[Xh

1 , Xh2

](q) = [X1, X2]h (q)−B(q)

(Xh

1 (q), Xh2 (q)

)q. (2.2.4)

2.3 Associated Bundles.

Besides the principal bundle π : Q → Q/G discussed above, consider a left actionρ : G ×M → M of the Lie group G on a manifold M . The associated bundlewith standard fiber M is, by definition,

Q×GM = (Q×M)/G,

where the action of G on Q×M is given by g(q,m) = (gq, gm). The class (or orbit)of (q,m) is denoted [q,m]G or simply [q,m]. The projection πM : Q×GM → Q/Gis defined by πM ([q,m]G) = π(q) and it is easy to check that it is well defined andis a surjective submersion.


Parallel Transport.

Let [q0,m0]G ∈ Q ×G M and let x0 = π(q0) ∈ Q/G. Let x(t), t ∈ [a, b] be a curvein Q/G and let t0 ∈ [a, b] be such that x(t0) = x0. The parallel transport of thiselement [q0,m0]G along the curve x(t) is defined to be the curve

[q,m]G(t) = [xhq0(t),m0]G.

Let us check that this curve is well defined. In fact, for any g ∈ G, the equivarianceproperty of the connection gives xhgq0(t) = gxhq0(t) for all t and hence

[xhgq0(t), gm0]G = [gxhq0(t), gm0]G = [xhq0(t),m0]G

for all t.Consider a curve x(t), t ∈ [a, b] in Q/G, as before. For t, t + s ∈ [a, b], we adopt

the notationτ tt+s : π−1

M (x(t))→ π−1M (x(t+ s))

for the parallel transport map along the curve x(s) of any point

[q(t),m(t)]G ∈ π−1M (x(t))

to the corresponding point

τ tt+s[q(t),m(t)]G ∈ π−1M (x(t+ s)) .

Thus,τ tt+s[q(t),m(t)]G = [xhq(t)(t+ s),m(t)]G.

Associated Vector Bundles.

Now we concentrate on the particular case when M is a vector space and ρ isa linear representation. (These will be the only associated bundles needed in thepresent work.) In this case, the associated bundle with standard fiber M is a vectorbundle in a natural way. We now recall what the vector bundle structure is. If[q,m]G, [q,m1]G, [q,m2]G ∈ π−1

M ([q]G), then the vector space structure in this fiberis defined by

a[q,m]G = [q, am]G and [q,m1]G + [q,m2]G = [q,m1 +m2]G.

We shall sometimes use the notation ρ′(ξ) for the second component of the in-finitesimal generator of an element ξ ∈ g, that is, ξm = (m, ρ′(ξ)m). Here we areusing the identification TM = M ×M , appropriate for vector spaces. Thus, we arethinking of the infinitesimal generator as a map ρ′ : g→ End(M) (the linear vectorfields on M are identified with the space of linear maps of M to itself). Thus, wehave a linear representation of the Lie algebra g on the vector space M .


Definition 2.3.1. Let [q(t),m(t)]G, t ∈ [a, b] be a curve in Q×GM , denote by

x(t) = πM ([q(t),m(t)]G) = π(q(t))

its projection on the base Q/G, and let, as above, τ tt+s, where t, t+s ∈ [a, b], denoteparallel transport along x(t) from time t to time t+ s. The covariant derivativeof [q(t),m(t)]G along x(t) is defined as follows

D[q(t),m(t)]GDt

= lims→0

τ t+st ([q(t+ s),m(t+ s)]G)− [q(t),m(t)]Gs

.

Thus, the covariant derivative of [q(t),m(t)]G is an element of π−1M (x(t)).

Notice that if [q(t),m(t)]G is a vertical curve, then its base point is constant; thatis, for each t ∈ [a, b],

x(t+ s) = πM ([q(t+ s),m(t+ s)]G) = x(t),

so that xhq(t)(t+ s) = q(t) for all s. Therefore,

τ t+st [q(t+ s),m(t+ s)]G = [xhq(t+s)(t),m(t+ s)]G = [q(t),m(t+ s)]G

and so we get the well known fact that the covariant derivative of a vertical curvein the associated bundle is just the fiber derivative. That is,

D[q(t),m(t)]GDt

= [q(t),m′(t)]G,

where m′(t) is the time derivative of m.

Affine Connections.

The notion of covariant derivative can be defined from a different, more axiomatic,point of view, which will be useful later in this paper. By definition (see Kobayashiand Nomizu (1963)), a connection (sometimes called an affine connection todistinguish it from a principal connection) ∇ on a vector bundle τ : V → Q is amap ∇ : X∞(Q) × Γ(V ) → Γ(V ), say (X, v) 7→ ∇Xv, having the following twoproperties:

1. First, we require that

∇f1X1+f2X2v = f1∇X1v + f2∇X2v

for all Xi ∈ X∞(Q) (the space of smooth vector fields on Q), fi ∈ C∞(Q)(the space of smooth real valued functions on Q), i = 1, 2, and all v ∈ Γ(V )(the space of smooth sections of the vector bundle V ),

2. and secondly,

∇X(f1v1 + f2v2) = X[f1]v1 + f1∇Xv1 +X[f2]v2 + f2∇Xv2

for all X ∈ X∞(Q), fi ∈ C∞(Q), and vi ∈ Γ(V ), i = 1, 2.


Here, X[f ] denotes the derivative of f in the direction of the vector field X.Given a connection on V , the parallel transport of a vector v0 ∈ τ−1(q0) along

a curve q(t) in Q, t ∈ [a, b] such that q(t0) = q0 for a fixed t0 ∈ [a, b], is the uniquecurve v(t) such that v(t) ∈ τ−1 (q(t)) for all t, v(t0) = v0, and which satisfies∇q(t)v(t) = 0 for all t. The operation of parallel transport establishes, as before, foreach t, s ∈ [a, b], a linear map

T tt+s : τ−1(q(t))→ τ−1(q(t+ s))

associated to each curve q(t) in Q. Then we can define the operation of covariantderivative on curves v(t) in V similar to that in the previous definition; that is,

Dv(t)Dt

=d

dsT t+st v(t+ s)

∣∣∣∣s=0

.

Observe that the connection ∇ can be recovered from the covariant derivative (andthus from the parallel transport operation). Indeed, ∇ is given by

∇Xv(q0) =D

Dtv(t)

∣∣∣∣t=t0

,

where, for each q0 ∈ Q, each X ∈ X∞(Q), and each v ∈ Γ(V ), we have, by definition,that q(t) is any curve in Q such that q(t0) = X(q0) and v(t) = v (q(t)) for all t. Thisproperty establishes, in particular, the uniqueness of the connection associated tothe covariant derivative D/Dt.

Formula for the Covariant Derivative Induced by a PrincipalConnection.

Now we return to the study of the affine connection induced on an associatedbundle. The following formula gives the relation between the covariant derivativeof the affine connection and the principal connection.

Lemma 2.3.2.

D[q(t),m(t)]GDt

= [q(t),−ρ′ (A (q(t), q(t)))m(t) + m(t)]G .

Proof. For fixed t and for any s we have

[q(t+ s),m(t+ s)]G = [gq(t+ s)qh(t+ s),m(t+ s)]G,

where we assume qh(t) = q(t) and gq(t) = e, the identity element of G. Then

τ t+st [q(t+ s),m(t+ s)]G = τ t+st [gq(t+ s)qh(t+ s),m(t+ s)]G= τ t+st [qh(t+ s), gq(t+ s)−1m(t+ s)]G= [qh(t), gq(t+ s)−1m(t+ s)]G.


Therefore, differentiating with respect to s at s = 0 we obtain

D[q(t),m(t)]GDt

= [q(t),−gq(t)m(t) + m(t)]G .

Since gq(t) = e we have, using Lemma 2.2.1, that gq(t) = A (q(t), q(t)). Therefore,

gq(t)m(t) = ρ′ (A (q(t), q(t)))m(t).

Induced Connections on Associated Bundles.

The previous definition of the covariant derivative of a curve in the associated vectorbundle Q×GM thus leads to a connection on Q×GM . Let us call this connection∇A or simply ∇.

We now describe this connection ∇A in more detail. Let ϕ : Q/G→ Q×GM be asection of the associated bundle and let X(x) ∈ Tx(Q/G) be a given vector tangentto Q/G at x. Let x(t) be a curve in Q/G such that x(0) = X(x); thus, ϕ (x(t)) isa curve in Q×GM . The covariant derivative of the section ϕ with respect to X atx is then, by definition,

∇AX(x)ϕ =Dϕ (x(t))

Dt

∣∣∣∣t=0

. (2.3.1)

Notice that we only need to know ϕ along the curve x(t) in order to calculate thecovariant derivative.

The notion of a horizontal curve [q(t),m(t)]G on Q ×G M is defined by thecondition that its covariant derivative vanishes. A vector tangent to Q×GM is calledhorizontal if it is tangent to a horizontal curve. Correspondingly, the horizontalspace at a point [q,m]G ∈ Q×GM is the space of all horizontal vectors at [q,m]G.

The Adjoint Bundle.

The case that interests us most in this paper occurs when M = g and ρg is theadjoint action Adg.

Definition 2.3.3. The associated bundle with standard fiber g, where the action ofG on g is the adjoint action, is called the adjoint bundle, and is sometimes denotedAd(Q). We will use the notation g := Ad(Q) in this paper. We let πG : g → Q/Gdenote the projection given by πG ([q, ξ]G) = [q]G.

Lemma 2.3.4. Let [q(s), ξ(s)]G be any curve in g. Then

D[q(s), ξ(s)]GDs

=[q(s),− [A (q(s), q(s)) , ξ(s)] + ξ(s)

]G.

Proof. Use the previous lemma and the fact that ρ′(ξ) = adξ.

The next Lemma says that the adjoint bundle is a Lie algebra bundle.


Lemma 2.3.5. Each fiber gx of g carries a natural Lie algebra structure definedby

[[q, ξ]G, [q, η]G] = [q, [ξ, η]]G .

Proof. We must show that the bracket is well defined, which is done in a straight-forward way as follows:

[[gq,Adg ξ]G, [gq,Adg η]G] = [gq, [Adg ξ,Adg η]]G= [gq,Adg[ξ, η]]G= [q, [ξ, η]]G= [[q, ξ]G, [q, η]G] .

2.4 The Bundles TQ/G and T (Q/G)⊕ g

Let π : Q → Q/G be a principal bundle with structure group G, as before. Thetangent lift of the action of G on Q defines an action of G on TQ and so we canform the quotient (TQ)/G =: TQ/G. There is a well defined map τQ/G : TQ/G→Q/G induced by the tangent of the projection map π : Q → Q/G and given by[vq]G 7→ [q]Q. The vector bundle structure of TQ is inherited by this bundle.

Lemma 2.4.1. The rules

[vq]G + [uq]G = [vq + uq]G and λ[vq]G = [λvq]G,

where λ ∈ R, vq, uq ∈ TqQ, and [vq]G and [uq]G are their equivalence classes in thequotient TQ/G, define a vector bundle structure on TQ/G having base Q/G. Thefiber (TQ/G)x is isomorphic, as a vector space, to TqQ, for each x = [q]G.

Proof. If [q0]G = x is given, the isomorphism between the fiber (TQ/G)xandTq0Q is given by the map [uq]G 7→ g−1uq, where g ∈ G is uniquely determined bythe relation gq0 = q.

The bundle TQ/G is a fundamental object in the present paper. One can statereduced variational principles in a natural way in terms of this bundle without anyreference to a connection on Q, which we shall describe in the next section.

It is, however, also interesting to introduce an (arbitrarily chosen) connection onQ relative to which one can realize the space TQ/G in a convenient way as well aswriting the Lagrange–Poincare equations in an interesting form. The next lemmais one of the main tools needed for doing this.

Lemma 2.4.2. The map αA : TQ/G→ T (Q/G)⊕ g defined by

αA ([q, q]G) = Tπ(q, q)⊕ [q, A(q, q)]G

is a well defined vector bundle isomorphism. The inverse of αA is given by

α−1A ((x, x)⊕ [q, ξ]G) = [(x, x)hq + ξq]G.

2.4 The Bundles TQ/G and T (Q/G)⊕ g 19

Proof. To show that αA is well defined, observe that for any g ∈ G we haveTπ(gq, gq) = Tπ(q, q) and also

[gq,A(gq, gq)]G = [gq,Adg A(q, q)]G = [q, A(q, q)]G.

Then we see thatαA ([gq, gq]G) = αA ([q, q]G) .

To show that α−1A is well defined, notice that (x, x)hgq = g(x, x)hq and that (Adg ξ)gq =

gξq. Therefore,

α−1A ((x, x)⊕ [gq,Adg ξ]G) = α−1

A ((x, x)⊕ [q, ξ]G) .

Remark.

The bundles TQ/G and T (Q/G) ⊕ g do not depend on the connection A, but, ofcourse, αA does.


Page 21

3The Lagrange–Poincare Equations

In this section we use the constructions from the previous section to show how towrite the Lagrange–Poincare equations in an intrinsic way. This is done in termsof an (arbitrarily chosen) connection A. The resulting equations are given on thebundle T (2)(Q/G)⊕ g, where T (2)(Q/G) is the second order tangent bundle (whichis related to the 2-jet bundle) of Q/G, whose general definition is recalled in §3.2,and where g = Q×G g is the associated adjoint bundle to Q. A key point in doingthis is to decompose arbitrary variations of curves in Q into vertical and horizon-tal components. This gives rise, correspondingly, to two reduced equations, namely,vertical Lagrange–Poincare equations, corresponding to vertical variations, and hor-izontal Lagrange–Poincare equations, corresponding to horizontal variations, whichare Euler–Lagrange equations on Q/G with an additional term involving the cur-vature B of A. (Our conventions for the curvature were given in §2.2)

3.1 The Geometry of Variations

Spaces of Curves.

The Setup. Fix a time interval I = [t0, t1]. The space of all (smooth) curves fromI to Q will be denoted Ω(Q). We shall not include the interval I explicitly in thenotation for spaces of curves for simplicity; it will be understood from the contextand explicitly stated when necessary.

Given a map f : Q1 → Q2, the map Ω(f) : Ω(Q1)→ Ω(Q2) is defined by

Ω(f)(q)(t) = f(q(t)),

22 3. The Lagrange–Poincare Equations

for q(t) an element of Ω(Q1). For given qi ∈ Q, i = 0, 1, by definition, Ω(Q; q0) andΩ(Q; q0, q1) are, respectively, the spaces of curves q(t) on Q such that q(t0) = q0

and q(ti) = qi, i = 0, 1.If π : Q → S is a bundle, q0 ∈ Q and π(q0) = x0, then Ω(Q;x0) denotes the

space of all curves in Ω(Q) such that π (q(t0)) = x0. The space Ω(Q;x1) is definedin an analogous way. Similarly, Ω(Q;x0, q1) is the space of all curves in Ω(Q) suchthat π (q(t0)) = x0 and q(t1) = q1. The spaces of curves Ω(Q; q0, x1), Ω(Q;x0, x1),etc. are defined in a similar way.

If V → Q and W → Q are vector bundles then Ω (V ) → Ω (Q) and Ω (W ) →Ω (Q) are vector bundles in a natural way and there is a natural identificationΩ (V ⊕W ) ≡ Ω(V )⊕ Ω(W ).

Deformations of Curves.

A deformation of a curve q(t) on a manifold Q is, by definition, a (smooth)function q(t, λ) such that q(t, 0) = q(t) for all t. The corresponding variation isdefined by

δq(t) =∂q(t, λ)∂λ

∣∣∣∣λ=0

.

Variations of curves q(t) belonging to Ω(Q; q0) or Ω(Q; q0, q1) satisfy the corre-sponding fixed endpoints conditions, namely, δq(t0) = 0 or δq(ti) = 0 for i = 0, 1,respectively.

Let τ : V → Q be a vector bundle and let v(t, λ) be a deformation in V of acurve v(t) in V . If τ (v(t, λ)) = q(t) does not depend on λ we will call v(t, λ) aV -fiber deformation of v(t), or simply, a fiber deformation of v(t). For each t,the variation

δv(t) =∂v(t, λ)∂λ

∣∣∣∣λ=0

may be naturally identified with an element, also called δv(t), of τ−1 (q(t)). In thiscase, the curve δv in V is, by definition, a V -fiber variation of the curve v, or,simply, a fiber variation of the curve v.

Horizontal and Vertical Variations.

We now break up the variation of a curve into horizontal and vertical parts. Thus,we consider a curve q ∈ Ω(Q; q0) (again, it is understood that the curves are definedon a fixed time interval [t0, t1]), where, as before, Q → Q/G is a principal bundlewith a connection A.

A vertical variation δq of q satisfies, by definition, the condition δq(t) =Ver(δq(t)) for all t. Similarly, a horizontal variation satisfies δq(t) = Hor (δq(t))for all t.

Clearly, any variation δq can be uniquely decomposed as follows:

δq(t) = Hor(δq(t)) + Ver(δq(t))

3.1 The Geometry of Variations 23

for all t, where Ver(δq(t)) = A(q(t), δq(t))q(t) and where Hor(δq(t)) = δq(t) −Ver(δq(t)).

Structure of Vertical Variations.

Given a curve q ∈ Ω(Q; q0, q1), let v = A(q, q) ∈ g. Variations δq of q(t) inducecorresponding variations δv ∈ g in the obvious way:

δv =∂A(q(t, λ), q(t, λ))

∂λ

∣∣∣∣λ=0

.

Consider the decomposition q = gqqh introduced in equation (2.2.2). A verticaldeformation q(t, λ) can be written as q(t, λ) = gq(t, λ)qh(t). The correspondingvariation δq(t) = δgq(t)qh(t) is of course also vertical.

Now we introduce some important notation. Define the curve

η(t) = δgq(t)gq(t)−1

in g. The fixed endpoint condition gives η(ti) = 0, i = 1, 2.Notice that, by construction,

δq(t) = δgq(t)qh(t) = η(t)gq(t)qh(t) = η(t)q(t).

Lemma 3.1.1. For any vertical variation δq = ηq of a curve q ∈ Ω(Q; q0, q1) thecorresponding variation δv of v = A(q, q) is given by δv = η + [η, v] with ηi = 0,i = 0, 1.

Proof. We will give the proof in the case that G is a matrix group. The moregeneral case can be treated using the appendix to Bloch, Krishnaprasad, Marsdenand Ratiu (1996).

By Lemma 2.2.1, we have v = gqg−1q . Then

δv = (δgq)g−1q − gqg−1

q δgqg−1q

= (δgq )g−1q − vη

= (ηgq + ηgq)g−1q − vη

= η + [η, v].

Note.

In the variational approach to the Euler–Poincare equations (see Marsden andRatiu (1999), Chapter 13), there is a class of constrained variations δξ = η +[ξ, η] introduced for computing the corresponding variational principle. The aboveconstruction of v, η is not computing the same objects. These constrained variations


are, instead, special instances of the construction of covariant variations, to beintroduced shortly in Definition 3.1.3. In the second remark following Lemma 3.1.4,we shall explicitly remark on how the constructions of variations for the Euler–Poincare equations and those for the Lagrange-Poincare case are related.

The Structure of Horizontal Variations.

Now we calculate variations δv corresponding to horizontal variations δq of a curveq ∈ Ω(Q; q0, q1).

Lemma 3.1.2. Let δq be a horizontal variation of a curve q ∈ Ω(Q; q0, q1). Thenthe corresponding variation δv of v = A(q, q) satisfies δv = B(q)(δq, q).

Proof. Let q(t, λ) be a horizontal deformation of q(t), that is, λ 7→ q(t, λ) is ahorizontal curve for each t. Now we work locally in a local trivialization of thebundle and write the connection A in the following way:

v = A(q, q) = 〈A(q), q〉 .

Then, we compute using the chain rule:

δv =∂v

∂λ

∣∣∣∣λ=0

=⟨DA(q) · ∂q

∂λ, q

⟩∣∣∣∣λ=0

+⟨A(q),

∂2q

∂λ∂t

⟩∣∣∣∣λ=0

.

On the other hand, since λ 7→ q(t, λ) is horizontal,⟨A(q),

∂q

∂λ

⟩= 0,

and so, by differentiating with respect to t,

0 =⟨DA(q) · q, ∂q

∂λ

⟩∣∣∣∣λ=0

+⟨A(q),

∂2q

∂t∂λ

⟩∣∣∣∣λ=0

.

Then we obtain, by subtraction,

δv = dA(q)(∂q

∂λ,∂q

∂t

)∣∣∣∣λ=0

.

Since ∂q/∂λ is horizontal, Cartan’s structure equation (2.2.1) implies

δv = B(q)(∂q

∂λ,∂q

∂t

)∣∣∣∣λ=0

or, in other words, δv = B(q)(δq, q).

3.1 The Geometry of Variations 25

The Covariant Variation on the Adjoint Bundle.

Any curve in Q, q ∈ Ω(Q; q0, q1) induces a curve in g in a natural way, namely,

[q, v]G(t) = [q(t), v(t)]G ,

where v(t) = A(q, q). Observe that, for each t, [q, v]G(t) ∈ gx(t) (the fiber over x(t)),where x(t) = π (q(t)) for all t. We want to study variations δ[q, v]G correspondingto vertical and also to horizontal variations δq of q.

While vertical variations δq give rise to vertical variations δ[q, v]G, horizontalvariations δq need not give rise to horizontal variations δ[q, v]G. The deviation ofany variation δ[q, v]G from being horizontal is measured by the covariant variationδA[q, v]G(t), defined as follows

Definition 3.1.3. For any given deformation q(t, λ) of q(t), the covariant vari-ation δA[q, v]G(t) is defined by

δA[q, v]G(t) =D [q(t, λ), v(t, λ)]G

Dλ

∣∣∣∣λ=0

.

Vertical Variations and the Adjoint Bundle.

We first consider the case of vertical variations.

Lemma 3.1.4. The covariant variation δA[q, v]G(t) corresponding to a verticalvariation δq = ηq is given by

δA[q, v]G(t) =D[q, η]GDt

+ [q, [v, η]]G .

Proof. Let q(t, λ) be a vertical deformation of q(t) with, as usual, δq = ∂q/∂λ|λ=0.As we saw before, q(t, λ) = gq(t, λ)qh(t) and δgq = ηgq, where η = δgqg

−1q =

A(q, δq). If we let v = A(q, q), by Lemma 3.1.1, we have δv = η + [η, v]. From thisand Lemma 2.3.4, we obtain

δA[q, v]G(t) = [q,−[A(q, δq), v] + δv]G= [q,−[η, v] + η + [η, v]]G= [q, η]G.

Again, by Lemma 2.3.4, we have

D[q, η]GDt

= [q,−[A(q, q), η] + η]G

= [q,−[v, η] + η]G= [q, η + [η, v]]G .

Therefore,

δA[q, v]G(t) =D[q, η]GDt

+ [q, [v, η]]G .


Remarks.

1. In view of Lemma 2.3.5, we can write

[q, [v, η]]G = [[q, v]G, [q, η]G] .

2. Let us now show that the formula δv = η + [v, η] for the constrained vari-ations for the Euler–Poincare equations (see Marsden and Ratiu (1999) andreferences therein) coincides with the construction of the covariant variationgiven in Definition 3.1.3. Given a Lie group G, we regard it as a principalbundle over a point; that is, we take G = Q. The identification of g withT (Q/G)⊕ g in this case is given by v 7→ [e, v]G. Then this equivalence definesδv ≡ δA[e, v]G and then the preceding lemma shows that δv = η+[v, η], whichis the same type of variation one has for the Euler–Poincare equations.

The Reduced Curvature Form.

In preparation for the consideration of variations δA[q, v]G(t) corresponding to hor-izontal variations, we prove the following result of independent interest.

Lemma 3.1.5. The curvature 2-form B ≡ BA of the connection A induces ag-valued 2-form B ≡ BA on Q/G given by

B(x)(δx, x) = [q,B(q)(δq, q)]G , (3.1.1)

where for each (x, x) and (x, δx) in Tx(Q/G), (q, q) and (q, δq) are any elements ofTqQ such that π(q) = x, Tπ(q, q) = (x, x) and Tπ(q, δq) = (x, δx).

Proof. We show that the right hand side does not depend on the choice of (q, q)and (q, δq). For any g ∈ G we have

[gq,B(gq)(gδq, gq)]G = [gq,Adg B(q)(δq, q)]G= [q,B(q)(δq, q)]G .

Definition 3.1.6. The g-valued 2-form B on Q/G will be called the reducedcurvature form.

Horizontal Variations and the Adjoint Bundle.

Now we are ready to describe covariant variations δA[q, v]G(t) corresponding tohorizontal variations δq.

Lemma 3.1.7. Variations δA[q, v]G(t) corresponding to horizontal variations δqare given by

δA[q, v]G(t) = B(x)(δx, x)(t),

where Tπ(q, q) = (x, x), Tπ(q, δq) = (x, δx), and v = A(q, q).

3.2 The Euler–Lagrange and Euler–Poincare Operators 27

Proof. By Lemma 2.3.4, we have δA[q, v]G(t) = [q,−[A(q, δq), v] + δv]G. Sinceδq is horizontal, we have A(q, δq) = 0. Using this and lemmas 3.1.2 and 3.1.5, weobtain δA[q, v]G(t) = B(x)(δx, x).

3.2 The Euler–Lagrange and Euler–PoincareOperators

The purpose of the next three sections is to carry out the reduction of the Euler–Lagrange equations by means of reduction of Hamilton’s principle using the ge-ometric set up in the preceding section. We will begin in this section with somegeometric preliminaries and treat the standard case of the Euler–Lagrange and theEuler–Poincare equations.

Reduced Spaces of Curves.

In what follows we shall often identify the bundles TQ/G and T (Q/G) ⊕ g, usingthe isomorphism αA of Lemma 2.4.2. This leads to other natural identificationsas well. For instance, the reduced set of curves [Ω(Q; q0, q1)]G is the set of curves[q]G(t) = [q(t)]G on Q/G such that the curve q(t) belongs to Ω(Q; q0, q1). Thisreduced set of curves is naturally identified with the set of curves [q(t), q(t)]G inTQ/G such that q(ti) = qi, for i = 0, 1, and in turn, this is identified, via the map

Ω(αA) : [Ω(Q; q0, q1)]G → Ω (T (Q/G)⊕ g) ,

with the set of curves

Tπ(q(t), q(t))⊕ [q(t), A (q(t), q(t))]G

in T (Q/G) ⊕ g, such that q(ti) = qi, for i = 0, 1. The image of this reduced set ofcurves will be denoted Ω(αA) ([Ω(Q; q0, q1)]G) .

The Reduced Lagrangian.

Let L : TQ → R be an invariant Lagrangian, that is, L (g(q, q)) = L(q, q) for all(q, q) ∈ TQ and all g ∈ G. Because of this invariance, we get a well defined reducedLagrangian l : TQ/G→ R satisfying

l ([q, q]G) = L(q, q).

As we will see in detail in this section, the evolution of the reduced system willbe a critical point, say a curve [q]G in the reduced set of curves [Ω(Q; q0, q1)]G, ofthe reduced action ∫ t1

t0

l ([q, q]G) dt


for suitable types of variations.However, variations of curves in the reduced family of curves are not of the usual

sort found in Hamilton’s principle, and so the equations of motion in the bundleTQ/G cannot be written in a direct way.

In this section we use the description of vertical and horizontal variations given inthe preceding section to derive equations of motion in the bundle T (2)(Q/G) ⊕ 2gdefined below. Equations corresponding to vertical variations will be called thevertical Lagrange–Poincare equations, and equations corresponding to horizontalvariations will be called the horizontal Lagrange–Poincare equations.

Identification of Bundles.

We shall allow a slight abuse of notation, namely we will consider l as a functiondefined on T (Q/G)⊕ g or TQ/G interchangeably, using the isomorphism αA. Alsowe shall often use a slight abuse of the variable-notation for a function, namely wewill write l(x, x, v) to emphasize the dependence of l on (x, x) ∈ T (Q/G) and v ∈ g.However one should keep in mind that x, x and v cannot be considered as beingindependent variables unless g and T (Q/G) are trivial bundles. Even in those casesin which g and T (Q/G) are trivial, and therefore x, x and v can be considered asbeing independent variables in a natural way, it is sometimes convenient to proceedusing the general theory.

The kth Order Tangent Bundle.

We define below the kth-order tangent bundle τ(k)Q : T (k)Q → Q. For q ∈ Q,

elements of T (k)q Q are equivalence classes of curves in Q, namely, two given curves

qi(t), i = 1, 2, such that q1(t1) = q2(t2) = q are equivalent, by definition, if and onlyif in any local chart we have q(l)

1 (t1) = q(l)2 (t2), for l = 1, 2, . . . , k, where q(l) denotes

the derivative of order l. The equivalence class of the curve q(t) at q = q(t) will bedenoted [q](k)

q . The projection

τ(k)Q : T (k)Q→ Q is given by τ

(k)Q

([q](k)q

)= q.

It is clear that T (0)Q = Q, T (1)Q = TQ, and that, for l < k, there is a welldefined fiber bundle structure

τ(l,k)Q : T (k)Q→ T (l)Q, given by τ

(l,k)Q

([q](k)q

)= [q](l)q .

The bundles T (k)Q for k > 1 are not vector bundles, except for k = 1. The bundleT (2)Q is often denoted Q, and is called the second order bundle (see, for example,Marsden, Patrick and Shkoller (1998), Marsden and Ratiu (1999) and referencestherein).


Relation to Jet Bundles.

Consider the bundle R × Q → R, whose sections are curves in Q (the fields ofclassical mechanics). Then the k-jet bundle of this bundle may be identified withthe bundle R × T (k)Q → R × Q, where the first component of this map is theidentity.

From the point of view of jet bundles associated to maps between two manifolds,T (k)Q coincides with the fiber bundle Jk0 (R, Q) formed by k-jets of curves from Rto Q (based at 0 ∈ R), as defined, for example in Bourbaki (1983) or Kolar, Michor,and Slovak (1993).

Properties of kth Order Tangent Bundles.

It is also easy to see that for any map f : M → N we have a naturally induced map

T (k)f : T (k)M → T (k)N given by T (k)f(

[q](k)q

)= [f q](k)

f(q).

In particular, a group action ρ : G × Q → Q can be naturally lifted to a groupaction

ρ(k) : G× T (k)Q→ T (k)Q given by ρ(k)g

([q](k)q

)= [ρg q](k)

ρ(g,q) .

We will often denote ρ(k)g

([q](k)q

)= ρ(k)

(g, [q](k)

q

)= g[q](k)

q .Let M ×N be the Cartesian product of the manifolds M and N . Then, for any

(m, n) ∈M×N there is a natural identification T (k)(m,n)((M×N) ≡ T (k)

m M×T (k)n N ,

which induces an identification T (k)(M ×N) ≡ T (k)M × T (k)N .The natural action of G on the fiber bundle T (k)Q endows T (k)Q with a principal

bundle structure with structure group G. The quotient T (k)Q/G can be easilyshown to be a fiber bundle over the base Q/G. The bundle T (2)Q/G is the onethat interests us most in this paper, because the Lagrange-Poincare operator of areduced Lagrangian is defined on T (2)Q/G with values in T ∗(Q/G)⊕ g∗, as we shallsee in §3.3. The class of the element [q](k)

q in the quotient T (k)Q/G will be denoted[[q](k)q

]G

, as usual. Since we have the projection πG(Q) : Q → Q/G we obtain abundle map

T (k)πG(Q) : T (k)Q→ T (k)(Q/G).

Moreover, it can be easily shown that this bundle map induces a well defined bundlemap

T (k)Q/G→ T (k)(Q/G) given by[[q](k)q

]G7→ T (k)πG(Q)

([q](k)q

).

Let q ∈ Q, denote π(q) = [q]G = x, and let [x](k)x ∈ T (k) (Q/G) be given. Let x(t)

be any curve belonging to the class [x](k)x . Then there is a unique horizontal lift xhq

of x(t). We define the horizontal lift of [x](k)x at q by

[x](k),hx,q :=

[xhq](k)

q.


We must also remark that T (k)G carries a natural Lie group structure.1 If [g](k)g ,

and [h](k)

hare classes of curves g and h in G, we define the product [g](k)

g [h](k)

has

being the class [gh](k)

ghat the point gh of the curve gh. The Lie algebra TeT (k)G of

T (k)G can be naturally identified, as a vector space, with (k+1)g (that is, the directsum of k + 1 copies of g), which, therefore, carries a unique Lie algebra structuresuch that this identification becomes a Lie algebra isomorphism. There is also anatural identification of T (k)

e G with kg.Also, for k = 1, 2, . . ., T (k)Q is a principal bundle with structure group T (k)G in

a natural way. More precisely, if [g](k)g ∈ T (k)G is the class of a curve g in G and

[q](k)q ∈ T (k)

q (Q) is the class of a curve q in Q we let [g](k)g [q](k)

q ∈ T (k)gq Q denote the

class [gq](k)gq of the curve gq at the point gq. In particular, if ξ ∈ kg and [q](k)

q ∈ T (k)Q

are given, there is a well defined element ξ[q](k)q ∈ T (k)Q.

Connection-like Structures on Higher Order TangentBundles.

For q ∈ Ω(Q), we have the curve [q(t), v(t)]G in g, where v(t) = A(q(t), q(t)). Lemma2.3.4 shows that the covariant derivative of [q(t), v(t)]G is given by

D[q(t), v(t)]GDt

= [q(t),− [A (q(t), q(t)) , v(t)] + v(t)]G

= [q(t), v(t)]G .

The second covariant derivative of [q(t), v(t)]G, again by Lemma 2.3.4, is given by

D2[q(t), v(t)]GDt2

= [q(t),− [v(t), v(t)] + v(t)]G .

More generally, for each k = 1, 2, . . ., we can find, by induction, a curve vk(t) in g,having an expression that involves v(t) and the derivatives v(l)(t), l = 1, 2, . . . , k−1,such that

Dk−1[q(t), v(t)]GDtk−1

= [q(t), vk(t)]G .

More precisely, we have

v1 = v and vk+1 = −[v, vk] + vk,

1Recall that T (1)G = TG is the semidirect product group Gsg whose Lie algebra is thesemidirect product gsg, where the second factor is regarded as the representation space of theadjoint action. This semidirect product Lie algebra is, as a vector space, equal to 2g := g⊕ g. Wewill not need, or study, the Lie group structure of T (k)G in this paper, although this would beinteresting to do. It would also be interesting to see if there is any relation between the structures

here and the study of the algebras that occur in the BBGKY hierarchy, as in Marsden, Morrison

and Weinstein (1984).


for k = 1, 2, . . .. In particular, we obtain

v2(t) = v(t), v3(t) = − [v(t), v(t)] + v(t),

etc. In addition, we shall write, by definition, v0(t) = 0. Using the fact that v(t) =A (q(t), q(t)), we can also find expressions for vk(t) in coordinates in terms of q(l)(t),l = 1, 2, . . . , k. We state the following lemma, which is readily proved.

Lemma 3.2.1. Let q(t) be a given curve in Q such that q(t) = q. For eachk = 1, 2, . . . the formula

Ak

([q](k)q

)= vk(t)

gives a well defined map Ak : T (k)Q→ g. Therefore there is also a well defined mapAk : T (k)Q→ kg, given, for each k = 1, 2, . . ., by

Ak

([q](k)q

)= ⊕kl=1vl(t),

where we have written kg to stand for the vector space direct sum ⊕kl=1g of k copiesof g.

Let g ∈ G and [q](k)q ∈ T (k)

q Q be given. Then we can easily prove that

Ak

(g[q](k)

q

)= Adg Ak

([q](k)q

),

using induction, the definition of Ak, and taking into account the formulas

Adg vk =d

dtAdg vk

and Adg[v, vk] = [Adg v,Adg vk] . According to Lemma 2.2.1, for any curve q(t) inQ such that q(t) = q we have

A(q, q) = gqg−1q .

Using this equation and the definition of vk, one can inductively find an expressionfor vk(t) in terms of g(l)

q (t), for l = 1, 2, . . .. For instance, for the case of matrixgroups, we can see directly that v1(t) = v(t) = gq(t), v2(t) = gq(t)− gq(t)2, etc. Itis not difficult to see that the expression for vk(t) is the sum of g(k)

q (t) plus termsinvolving only the lower order derivatives g(l)

q (t), l = 1, 2, . . . , k− 1. Using this fact,one sees that for any element ξ = (ξ1, . . . ξk) ∈ kg and any element q ∈ Q there isa unique point [q](k)

q ∈ T (k)q Q which equals [gq](k)

q for a curve g(t) in G satisfying

g(t) = e and Ak

([q](k)q

)= ξ.

In fact, since we obviously have gq(t) = g(t), it is enough to find g(t) such that thederivatives g(l)(t), l = 1, 2, . . . , k, satisfy the appropriate conditions as explained


above. We shall call this unique element ξq, as before, and the set of all suchelements will be denoted kgq. Moreover, it is not difficult to see that the restrictionAk : kgq → kg is a diffeomorphism and therefore it naturally defines a uniquevector space structure on kgq such that the restriction of Ak becomes a linearisomorphism Ak(ξq) = ξ, for all xi ∈ kg. By construction, we see that there is anatural identification between kgq and T (k)

q (Gq). Note that we have Ak(ξq) = ξ forall ξ ∈ kg, analogous to what one has for connections.

Let us define, for each k = 1, 2, . . ., the vector bundle kg as being the Whitneysum of k copies of g. Define a map

T (k)Q→ kg by [q](k)q 7→

[q, Ak

([q](k)q

)]G,

where the last term is defined by[q, Ak

([q](k)q

)]G

= ⊕kl=1

[q, Al

([q](l)q

)]G.

The definitions show that, given any curve q(t) in Q such that q(t) = q, we have,at t = t, [

q, Ak

([q](k)q

)]G

=k⊕l=1

D(l−1)[q(t), v(t)]GDt(l−1)

∣∣∣∣∣t=t

.

We have essentially proven the following lemma, which generalizes Lemma 2.4.2

Lemma 3.2.2. The map

αAk : T (k)Q/G→ T (k)(Q/G)×Q/G kg

defined by

αAk

([[q](k)q

]G

)= T (k)πG(Q)

([q](k)q

)×Q/G

[q, Ak

([q](k)q

)]G

is a well defined bundle isomorphism. The inverse of αAk is given by

α−1Ak

([x](k)

x ×Q/G [q, ξ]G)

= ξ[x](k),hx,q .

Remark. As we have said earlier, the bundles T (k)Q, for k = 1, 2, are the onlyones that interest us in this paper. The cases k = 2, 3, . . . are needed, for instance,to deal with higher order Lagrangians L : T (k)Q → R. Then the Euler–Lagrangeoperator EL(L) will be defined on T (k+1)Q and will take values in a special vectorbundle. There are also several interesting structures on the bundles T (k)Q whichwe will not study in the present paper.


Euler–Lagrange Operator.

Next we introduce some notation and recall some basic results concerning Euler–Lagrange operators. The fundamental connection between the variational and diff-erential-equation description of the evolution of a given system is given by thefollowing well known result.

Theorem 3.2.3 (Euler–Lagrange). Let L : TQ→ R be a given Lagrangian on amanifold Q and let

S(L)(q) =∫ t1

t0

L(q, q) dt

be the action of L defined on Ω(Q; q0, q1). Let q(t, λ) be a deformation of a curveq(t) in Ω(Q; q0, q1) and let δq(t) be the corresponding variation. Then, by definition,δq(ti) = 0 for i = 0, 1.

There is a unique bundle map

EL(L) : T (2)Q→ T ∗Q

such that, for any deformation q(t, λ), keeping the endpoints fixed, we have

dS(L)(q) · δq =∫ t1

t0

EL(L)(q, q, q) · δq,

where, as usual,

dS(L)(q) · δq =d

dλS(L) (q(t, λ))

∣∣∣∣λ=0

with

δq(t) =∂q(t, λ)∂λ

∣∣∣∣λ=0

.

The 1-form bundle-valued map EL(L) is called the Euler–Lagrange operator.

In local coordinates EL(L) has the following well known expression:

EL(L)i(q, q, q) dqi =(∂L

∂qi(q, q)− d

dt

∂L

∂qi(q, q)

)dqi

in which it is understood that one regards the second term on the right hand sideas a function on the second order tangent bundle by formally applying the chainrule and then replacing everywhere dq/dt by q and dq/dt by q. The Euler–Lagrangeequations can, of course, be written simply as EL(L)(q, q, q) = 0.

Euler–Poincare Operator.

Analogous to the Euler–Lagrange operator, the Euler–Poincare theorem (Marsdenand Scheurle (1993b) and Bloch, Krishnaprasad, Marsden and Ratiu (1996); see alsoMarsden and Ratiu (1999), section 13.5) induces an operator, called the Euler–Poincare operator. It is defined, as before, by the variational principle.


Theorem 3.2.4 (Euler–Poincare). Let G be a Lie group, L : TG → R a leftG-invariant Lagrangian,

S(L)(g) =∫ t1

t0

L(g, g) dt

the action functional of L defined on Ω(G; g0, g1), l = L|g the reduced Lagrangian,and

Sred(l)(v) =∫ t1

t0

l(v) dt

the reduced action functional defined on Ω(g). Let g(t, λ) be a deformationof a curve g(t) in Ω(G; g0, g1), keeping the endpoints fixed, and let δg(t) be thecorresponding variation; thus, by definition, δg(ti) = 0, for i = 0, 1. Let v(t) =g(t)−1g(t) ∈ g.

The following are equivalent:

(i) the curve g(t) satisfies the Euler–Lagrange equations EL(L)(g, g, g) = 0 on G;

(ii) the curve g(t) is a critical point of the action functional S(L) for variationsδg vanishing at the endpoints;

(iii) the curve v(t) solves the Euler-Poincare equations

d

dt

∂l

∂v= ad∗v

∂l

∂v.

(iv) the curve v(t) is a critical point of the reduced action functional

Sred(l)(v) =∫ t1

t0

l(v(t))dt,

for variations of the formδv = η + [v, η],

where η(t) ∈ g is an arbitrary curve that vanishes at the endpoints. Thesevariations δv are exactly the variations induced by left translation of arbitrarydeformations g(t, λ) of the curve g(t) = g(t, λ) such that δg(ti) = 0, fori = 0, 1.

In addition, there is a unique bundle map

EP(l) : 2g→ g∗

where 2g := g⊕ g (in accordance with the definition introduced in the statement ofLemma 3.2.1), such that, for any deformation v(t, λ) = g(t, λ)−1g(t, λ) ∈ g inducedon g by a deformation g(t, λ) ∈ G of g(t) ∈ Ω(G; g0, g1) keeping the endpoints fixed,and thus δg(ti) = 0, for i = 0, 1, we have

dSred(l)(v) · δv =∫ t1

t0

EP(l)(v, v) · η dt,

3.3 The Lagrange–Poincare Operator 35

where, as usual,

dSred(l)(v) · δv =d

dλSred(l) (v(t, λ))

∣∣∣∣λ=0

and δv(t) = ∂v(t, λ)/∂t|λ=0 = η(t) + [v(t), η(t)].The map EP(l) is called the Euler–Poincare operator and its expression is

given by

EP(l)(v, v) = ad∗vδl

δv− d

dt

δl

δv

where, as before, it is to be understood that the time derivative on the second termis performed formally using the chain rule and that the expression dv/dt is replacedthroughout by v.

The Euler–Poincare equations can be written simply as EP(l)(v, v) = 0. Theformula δv = η + [v, η] represents the most general variation δv of v induced byan arbitrary variation δg via left translation. The precise relationship is η = g−1δgand so the condition δg = 0 at the endpoints is equivalent to the condition η = 0at the endpoints.

3.3 The Lagrange–Poincare Operator

In this section we introduce the Lagrange-Poincare operator using the same typeof technique of reduction of variational principles that was used in the precedingsection to define the Euler–Lagrange and the Euler–Poincare operators.

Reducing the Euler–Lagrange Operator.

The map EL(L) : T (2)Q→ T ∗Q, being G- equivariant, induces a quotient map

[EL(L)]G : T (2)Q/G→ T ∗Q/G,

which depends only on the reduced Lagrangian l : TQ/G→ R; that is, we can iden-tify [EL(L)]G with an operator EL(l). This is called the reduced Euler–Lagrangeoperator and it does not depend on any extra structure on the principal bundleQ. However, to write the explicit expressions, which are also physically meaningful,we use the additional structure of a principal connection A on the principal bundleQ→ Q/G to identify the quotient bundle

T (2)Q/G with T (2)(Q/G)×Q/G 2g

andT ∗Q/G with T ∗(Q/G)⊕ g∗


using the bundle isomorphisms αA2 from Lemma 3.2.2 and αA from Lemma 2.4.2,and also a connection ∇ on Q/G to concretely realize the reduced Euler–Lagrangeoperator; this will naturally lead us to the Lagrange–Poincare operator.2

Geometry of Reduced Variations.

A general variation δv(t) of a given curve v(t) in g is constructed as follows: choosea family of curves v(t, s) in g such that v(t, 0) = v(t) and define

δv(t) =∂v(t, s)∂s

∣∣∣∣s=0

.

This δv(t) is, for each t, an element of T g. However, it turns out that we will notneed these kinds of general variations δv subsequently. Instead, we are interestedin the special kind of deformations v(t, s) of the curve v(t) in which the projectionπG (v(t, s)) = x(t, s) does not depend on s, that is, deformations that take place onlyin the fiber of g over x(t) = πG (v(t)); thus, for each fixed t, the curve s 7→ v(t, s)is a curve in the fiber over x(t). Then, since g is a vector bundle, the variationδv(t) induced by such a deformation v(t, s), is naturally identified with a curve,also called δv(t), in g, a g-fiber variation, according to the notation introducedin the paragraph Deformation of Curves of §3.1. We remark that, in the restof this paper, δv will always mean a g-variation, unless explicitly stated otherwise.Also, the meaning of δv as an element of T g will never be used without an explicitprevious warning.

Examples of g-fiber variations δv are, for instance, the covariant variations δAvconsidered in Definition 3.1.3, but, of course, there are more general variations ofthis type. We have already encountered an easy example of such variations whenreviewing the Euler–Poincare equations. In that case, Q = G, the connection A :TG → g is given by right translation, and δAv(t) = η(t) + [v(t), η(t)], for η(t) acurve in g vanishing at the endpoints.

Of course, in the Euler–Poincare case, it is obviously true that any deformationof a curve v(t) is a deformation along the fiber, because the base of g is a point.However, as we have seen in the Euler–Poincare Theorem 3.2.4, it is not true thatany curve in g is induced by a variation δg that vanishes at the endpoints; thelatter are only the curves of the type η(t) + [v(t), η(t)], for η(t) an arbitrary curvein g vanishing at the endpoints.

In the study of the Lagrange-Poincare operator and Lagrange-Poincare equationwe will use variations of curves in Q/G⊕ g. (As explained in paragraph Bundles of§2.1, the first summand means the vector bundle over Q/G with zero dimensionalfiber). For a given curve x(t)⊕ v(t) in Q/G⊕ g, and a given arbitrary deformationx(t, λ)⊕ v(t, λ), with x(t, 0)⊕ v(t, 0) = x(t)⊕ v(t), of it, the corresponding covariant

2In what follows we will assume that the connection ∇ is chosen to be torsion free. For an

account of the situation when a connection with torsion is chosen, see Gamboa Saravı and Solomin(2003).


variation δx(t)⊕ δAv(t) is, by definition,

δx(t)⊕ δAv(t) =∂x(t, s)∂s

∣∣∣∣s=0

⊕ Dv(t, s)Ds

∣∣∣∣s=0

.

It is clear that δAv is a g-fiber variation of v. The most important example ofa covariant variation δx(t) ⊕ δAv(t) is the one to be described next. Let q(t, s)be a deformation of a curve q(t) = q(t, 0) in Q. This induces a deformationx(t, s) ⊕ v(t, s) of the curve x(t) ⊕ v(t) by taking x(t, s) = [q(t, s)]G and v(t, s) =[q(t, s), A(q(t, s), q(t, s))]G, where q(t, s) represents the derivative with respect tot. Using Lemma 2.3.4 and Definition 3.1.3, it follows that the covariant variationcorresponding to this deformation of x(t)⊕ v(t) is δx(t)⊕ δAv(t), where

δAv(t) =D[q(t), η(t)]G

Dt+ [q(t), [A(q(t), q(t)), η(t)]]G + B(δx(t), x(t)),

is an element of g for each t, with η(t) ∈ g an arbitrary curve vanishing at theendpoints. This is a special kind of covariant variation. It is precisely to these kindsof variations that we will apply the usual techniques of the calculus of variationsin the next theorems to derive the Lagrange-Poincare operator and equation. Theprevious formula may be rewritten as follows, which emphasizes the similarity withthe Euler-Poincare case,

δAv(t) =Dη

Dt(t) + [v(t), η(t)] + B(δx(t), x(t)),

where η = [q(t), η(t)]G.

Lagrange-Poincare Operator.

We are now ready to state a theorem that introduces the Lagrange-Poincare oper-ator. Its proof will be contained in the proof of Theorem 3.3.4.

Theorem 3.3.1. Let L : TQ → R be an invariant Lagrangian on the principalbundle Q as before. Choose a principal connection A on Q and identify the bundlesTQ/G and T (Q/G) ⊕ g using the isomorphism αA and also the bundles T (2)Q/Gand T (2)(Q/G) ×Q/G 2g using the isomorphism αA2 , as before. Thus an element[q, q]G of TQ/G can be written, equivalently, as an element (x, x, v) of T (Q/G)⊕ g.Let l : T (Q/G)⊕ g→ R be the reduced Lagrangian. Then there is a unique bundlemap

LP(l) : T (2)(Q/G)×Q/G 2g→ T ∗(Q/G)⊕ g∗

such that for any curve q ∈ Ω(Q; q0, q1) and any variation δq of q vanishingat the endpoints, the corresponding reduced curve [q, q]G = (x, x, v), where v =[q, A(q, q)]G, and covariant variation δx⊕ δAv, where

δAv(t) =Dη

Dt(t) + [v(t), η(t)] + B(δx(t), x(t)),


with η(t) = [q(t), η(t)]G and

δx(t) = Tπ(δq(t)),

satisfy

EL(L)(q(t), q(t), q(t)) · δq(t) = LP(l)(x(t), x(t), v(t)) · (δx(t)⊕ η(t)).

Notice that, after all the identifications described at the beginning of the presentparagraph, the operator LP(l) coincides with the operator [EL(L)]G.

Definition 3.3.2. The 1-form valued bundle map

LP(l) : T (2)(Q/G) ×Q/G 2g→ T ∗(Q/G)⊕ g∗

defined in the preceding theorem will be called the Lagrange–Poincare operator.The decomposition of the range space for LP(l) as a direct sum naturally inducesa decomposition of the Lagrange-Poincare operator

LP(l) = Hor(LP)(l)⊕Ver(LP)(l)

which define the horizontal Lagrange–Poincare operator and the vertical La-grange–Poincare operator.

The Lagrange–Poincare equations are, by definition, the equations LP(l) =0. The horizontal Lagrange–Poincare equation and vertical Lagrange–Poincareequation are, respectively, the equations

Hor(LP)(l) = 0 and Ver(LP)(l) = 0.

In the following paragraph we introduce some additional structure, namely, anarbitrary connection ∇ on the manifold Q/G. This will also help us write explicitexpressions of Hor(LP)(l) and Ver(LP)(l).

The problem of computing the Lagrange–Poincare equations can be done usingany connection, as we remarked earlier and, in addition, the problem can be local-ized to any local trivialization of the bundle Q → Q/G. Because of this, one maychoose the vector space or trivial connection associated with such a local trivializa-tion of the bundle. Of course we are not assuming that the bundle has a global flatconnection.

Explicit formulas for Hor(LP)(l) and Ver(LP)(l) in coordinates using any con-nection can be calculated from what we have developed and are given in §4.2. Doingso, one arrives at the coordinate formulas given in Marsden and Scheurle (1993b).We also mention that it is possible to derive these equations from Cendra, Ibortand Marsden (1987) in a straightforward way.

Reduced Covariant Derivatives.

The question of calculating formulas for Hor(LP)(l) and Ver(LP)(l) rests on givingmeaning to the partial derivatives

∂l

∂x,

∂l

∂xand

∂l

∂v.


Since g and T (Q/G) are vector bundles, we may interpret the last two derivativesin a standard (fiber derivative) way as being elements of the dual bundles T ∗(Q/G)and g∗, for each choice of (x, x, v) in T (Q/G)⊕g. In other words, for given (x0, x0, v0)and (x0, x

′, v′) we define

∂l

∂x(x0, x0, v0) · x′ =

d

ds

∣∣∣∣s=0

l(x0, x0 + sx′, v0)

and∂l

∂v(x0, x0, v0) · v′ =

d

ds

∣∣∣∣s=0

l(x0, x0, v0 + sv′).

To define the derivative ∂l/∂x, one uses the chosen connection ∇ on the manifoldQ/G, as we will explain next. Let (x0, x0, v0) be a given element of T (Q/G) ⊕ g.For any given curve x(s) on Q/G, let (x(s), v(s)) be the horizontal lift of x(s) withrespect to the connection ∇A on g (see (2.3.1)) such that (x(0), v(0)) = (x0, v0) andlet (x(s), u(s)) be the horizontal lift of x(s) with respect to the connection ∇ suchthat (x(0), u(0)) = (x0, x0). (Notice that, in general, (x(s), u(s)) is not the tangentvector (x(s), x(s)) to x(s).)

Thus, (x(s), u(s), v(s)) is a horizontal curve with respect to the connection C =∇⊕∇A naturally defined on T (Q/G)⊕ g in terms of the connection ∇ on T (Q/G)and the connection ∇A on g.

Definition 3.3.3. The covariant derivative of l with respect to x at (x0, x0, v0)in the direction of (x(0), x(0)) is defined by

∂C l

∂x(x0, x0, v0) (x(0), x(0)) =

d

ds

∣∣∣∣s=0

l (x(s), u(s), v(s)) .

We shall often write∂C l

∂x≡ ∂l

∂x,

whenever there is no danger of confusion.

The covariant derivative on a given vector bundle, for instance g, induces a cor-responding covariant derivative on the dual bundle, in our case g∗. More precisely,let α(t) be a curve in g∗. We define the covariant derivative of α(t) in such a waythat for any curve v(t) on g, such that both α(t) and v(t) project on the same curvex(t) on Q/G, we have

d

dt〈α(t), v(t)〉 =

⟨Dα(t)Dt

, v(t)⟩

+⟨α(t),

Dv(t)Dt

⟩.

Likewise we can define the covariant derivative in the vector bundle T ∗(Q/G). Thenwe obtain a covariant derivative on the vector bundle T ∗(Q/G)⊕ g∗.

It is in the sense of this definition that terms like

D

Dt

∂l

∂x


in the second equation (which defines the horizontal Lagrange–Poincare operator)and

D

Dt

∂l

∂v

in the first equation (which defines the vertical Lagrange–Poincare equation) of thefollowing theorem should be interpreted. In this case D/Dt means the covariantderivative in the bundle T ∗(Q/G). In the first equation D/Dt is the covariantderivative in the bundle g∗.

Reduced Variational Principles & the Lagrange–PoincareEquations.

The main result in this section is the following theorem. Its proof also contains theproof of the preceding theorem.

Theorem 3.3.4. Assume the hypothesis of Theorem 3.3.1 Then:the vertical Lagrange–Poincare operator is given by

Ver(LP)(l) · η =(− D

Dt

∂l

∂v(x, x, v) + ad∗v

∂l

∂v(x, x, v)

)· η

or simply,

Ver(LP)(l) =(− D

Dt

∂l

∂v(x, x, v) + ad∗v

∂l

∂v(x, x, v)

)and the horizontal Lagrange–Poincare operator is given by

Hor(LP)(l) · δx =(∂l

∂x(x, x, v)− D

Dt

∂l

∂x(x, x, v)

)δx− ∂l

∂v(x, x, v)B(x)(x, δx),

or simply,

Hor(LP)(l) =(∂l

∂x(x, x, v)− D

Dt

∂l

∂x(x, x, v)

)− ∂l

∂v(x, x, v)B(x)(x, .).

Proof. To compute the vertical and horizontal Lagrange-Poincare operator, itsuffices to consider variations δAv of a curve x(t) ⊕ v(t) corresponding to verticaland horizontal variations δq of a curve q ∈ Ω(Q, q0, q1). The computations belowwill show that these variations suffice to give us the variational principle in thedirections of the two summands in δx⊕ η ∈ T (Q/G)⊕ g.

First, consider variations δAv of a curve x(t) ⊕ v(t) corresponding to verticalvariations δq of a curve q. We have

0 = δ

∫ t1

t0

l(x, x, v)dt =∫ t1

t0

∂l

∂v(x, x, v)δAvdt.

3.4 The Reduced Variational Principle 41

According to Lemma 3.1.4 with v = [q, v]G we obtain, for all curves η(t) ∈ g suchthat η(ti) = 0 for i = 1, 2, the equation

0 =∫ t1

t0

⟨∂l

∂v,D[q, η]GDt

+ [q, [v, η]]G

⟩dt

=∫ t1

t0

⟨− D

Dt

∂l

∂v+ ad∗v

∂l

∂v, [q, η]G

⟩dt.

Arbitrariness of η then yields arbitrariness of η = [q, η]G, so we get

Ver(LP)(l) = − D

Dt

∂l

∂v(x, x, v) + ad∗v

∂l

∂v(x, x, v).

Now consider variations δx ⊕ δAv corresponding to horizontal variations δq. Thenwe have, for all δx with δx(ti) = 0, for i = 0, 1

δ

∫ t1

t0

l(x, x, v)dt =∫ t1

t0

(∂l

∂xδx+

∂l

∂xδx+

∂l

∂vδAv

)dt.

Integration by parts and Lemma 3.1.7 with v = [q, v]G gives

δ

∫ t1

t0

l(x, x, v)dt =∫ t1

t0

[(∂l

∂x− D

Dt

∂l

∂x

)(x, x, v)δx− ∂l

∂v(x, x, v)B(x)(x, δx)

]dt.

Integration by parts of the term (∂l/∂x)δx is justified by showing that

δx =D

Dλ

∂x

∂t=

D

Dt

∂x

∂λ,

which can be done, for example, by using Gaussian coordinates relative to theconnection ∇ at each point x(t). Arbitrariness of δx then yields

Hor(LP)(l)(x, x, v) =∂l

∂x(x, x, v)− D

Dt

∂l

∂x(x, x, v)− ∂l

∂v(x, x, v)ixB(x).

3.4 The Reduced Variational Principle

Now we turn to the reduction of the variational principle. As we stated at thebeginning of this section, Hamilton’s principle for a G-invariant L is equivalentto a reduced variational principle for l with respect to a reduced set of curves[Ω(Q; q0, q1)]G. Translated to the concrete realizations of our reduced bundles usingthe map αA, this reads as follows.

The following conditions are equivalent:


(i) Hamilton’s principle holds: the curve q(t) is a critical point of the actionfunctional ∫ t1

t0

L(q, q)dt

on Ω(Q; q0, q1);

(ii) the reduced variational principle holds: the curve (x(t), v(t)) is a criticalpoint of the action functional∫ t1

t0

l (x(t), x(t), v(t)) dt

on the reduced family of curves Ω (αA ([Ω(Q; q0, q1)]G)).

Now comes a main theorem which summarizes what we have done so far.

Theorem 3.4.1. The following conditions are equivalent:

(i) Hamilton’s principle holds: the curve q(t) is a critical point of the actionfunctional ∫ t1

t0

L(q, q)dt

on Ω(Q; q0, q1), that is,

δ

∫ t1

t0

L(q, q)dt = 0

for arbitrary variations δq of the curve q such that δq(ti) = 0, for i = 0, 1.

(ii) The reduced variational principle holds: the curve x(t)⊕ v(t) is a criticalpoint of the action functional∫ t1

t0

l (x(t), x(t), v(t)) dt

on the reduced family of curves αA ([Ω(Q; q0, q1)]G), that is,

δ

∫ t1

t0

l (x(t), x(t), v(t)) dt = 0,

for variations δx⊕ δAv of the curve x(t)⊕ v(t), where δAv has the form

δAv =Dη

Dt+ [v, η] + B(δx, x),

with the boundary conditions δx(ti) = 0 and η(ti) = 0, for i = 0, 1. If v =[q, v]G with v = A(q, q) then η can be always written η = [q, η]G, and thecondition η(ti) = 0 for i = 0, 1 is equivalent to the condition η(ti) = 0 fori = 0, 1.

3.4 The Reduced Variational Principle 43

Also, if x(t) = [q]G and v = [q, v]G where v = A (q, q), then variations δx⊕δAvsuch that

δAv =Dη

Dt+ [v, η]

≡ D[q, η]GDt

+ [q, [v, η]]G

with η(ti) = 0 (or, equivalently, η(ti) = 0) for i = 0, 1 correspond exactly tovertical variations δq of the curve q such that δq(ti) = 0 for i = 0, 1, whilevariations δx⊕ δAv such that

δAv = B(δx, x)

with δx(ti) = 0 for 1 = 0, 1, correspond exactly to horizontal variations δq ofthe curve q such that δq(ti) = 0.

(iii) The following vertical Lagrange–Poincare equations, corresponding tovertical variations, hold:

D

Dt

∂l

∂v(x, x, v) = ad∗v

∂l

∂v(x, x, v)

and the horizontal Lagrange–Poincare equations, corresponding to hor-izontal variations, hold:

∂l

∂x(x, x, v)− D

Dt

∂l

∂x(x, x, v) =

⟨∂l

∂v(x, x, v), ixB(x)

⟩.

Remarks.

1. The operators EL(l), Hor(LP)(l) and Ver(LP)(l) depend on the (principal)connection A on the principal bundle Q but not on the connection ∇ onQ/G. It is only the explicit expressions of Hor(LP)(l) and Ver(LP)(l) thatappear in Theorem 3.3.4 that depend on ∇. As we have remarked previously,in local coordinates it is often convenient to choose ∇ to be simply the usualEuclidean, or vector space connection.

2. Important particular cases of Theorems 3.3.1 and 3.3.4 occur when G =Q and also when G = e. If G = Q then the operator Hor(LP)(l) is 0and Ver(LP)(l) is the Euler–Poincare operator, as we saw before. Thus, in asense, the vertical Lagrange–Poincare operator in the bundle g is a covariantversion of the usual Euler–Poincare operator on a Lie algebra. If G = e thenVer(LP)(l) is 0, L = l and Hor(LP)(l) = EL(L) is the usual Euler–Lagrangeoperator.


Page 45

4Wong’s Equations and CoordinateFormulas

To illustrate the Lagrange-Poincare theory that we have developed, we first consideran interesting example, that of Wong’s equations. Secondly, in this section we givecoordinate expressions for the Lagrange-Poincare equations. Wong’s equations arefirst done intrinsically and then are used to illustrate the coordinate formulas.

4.1 Wong’s Equations

Context of Wong’s Equations.

Wong’s equations arise in at least two different interesting contexts. The first ofthese, in the work of Wong (1970), Sternberg (1977), Weinstein (1978) and Mont-gomery (1984), concerns the dynamics of a colored particle in a Yang-Mills field.The second context is that of the falling cat theorem of Montgomery [1990, 1993].For a direct proof of the falling cat theorem using the ideas of Lagrangian reduction,see Koon and Marsden (1997a) and Cendra, Holm, Marsden and Ratiu (1998).

Abstract Setting.

Let (X, g) be a given Riemannian manifold and let ∇ be the corresponding Levi-Civita connection. Let G be a compact Lie group with a bi-invariant Riemannianmetric κ. Let π : Q → X be a principal bundle with structure group G acting onthe left, let A be a principal connection on Q, and let B be the curvature of A.

46 4. Wong’s Equations and Coordinate Formulas

Now define the Lagrangian L : TQ→ R by

L(q, q) =12κ (A(q, q), A(q, q)) +

12g (π(q)) (Tπ(q, q), Tπ(q, q)) .

This Lagrangian is G-invariant and our object is to carry out the constructions forLagrangian reduction as described in the preceding sections to this situation.

We note that in the special case of G = S1, this Lagrangian is the Kaluza-Klein Lagrangian for the motion of a particle in a magnetic field. In this case, theconstructions are done directly in Marsden and Ratiu (1999). More generally, thisLagrangian is the Kaluza-Klein Lagrangian for particles in a Yang-Mills field A.

Construction of the Reduced Bundle.

An element of g has the form v = [q, v]G where q ∈ Q and v ∈ g. Since κ is bi-invariant, its restriction to g is Ad-invariant, and so we can define the fiber metrick on g by

k ([q, u]G, [q, v]G) = κ(u, v).

The reduced bundle is T (Q/G)⊕ g ≡ TX⊕ g and a typical element of it is denoted(x, x, v). The reduced Lagrangian is given by

l(x, x, v) =12k(v, v) +

12g(x)(x, x).

Calculation of the Reduced Equations.

Now we will write the vertical and horizontal Lagrange–Poincare equations. Westart by writing the vertical Lagrange-Poincare equation from Theorem 3.3.1 asfollows: (

− D

Dt

∂l

∂v(x, x, v) + ad∗v

∂l

∂v(x, x, v)

)· η = 0 (4.1.1)

for all η ∈ g. We first note that

∂l

∂v(x, x, v) = k(v, ·)

and hence (ad∗v

∂l

∂v(x, x, v)

)η = k (v, [v, η]) = 0,

since κ and hence k are bi-invariant. Thus, the vertical Lagrange–Poincare equationis

D

Dtk(v, ·) = 0,

which is one of Wong’s equations, namely the charge equation. We will see thisexplicitly in coordinates later.

4.2 The Local Vertical and Horizontal Equations 47

From Theorem 3.4.1, the horizontal Lagrange-Poincare equation is

∂l

∂x(x, x, v)− D

Dt

∂l

∂x(x, x, v) =

⟨∂l

∂v(x, x, v), ixB(x)

⟩.

Perhaps the easiest way to work out this expression is to do so in a local trivializationof the principal bundle, which induces a corresponding trivialization of g. In sucha local trivialization, the metric k is independent of the base point x. Making useof the vertical equation, the left hand side of the preceding equation becomes theusual Euler–Lagrange expression. Note that this expression is independent of whichaffine connection is used on X. It is well known that the Euler–Lagrange expressionfor the kinetic energy on X gives the covariant acceleration ∇xx using the Levi-Civita connection for the metric on X. Therefore, the horizontal Lagrange–Poincareequation becomes

(∇xx)[ = −k(v, B(x)(x, ·)

),

which is the second Wong equation.

4.2 The Local Vertical and Horizontal Equations

In this section we shall derive local formulas (that is, for a local trivialization ofthe principal bundle) of both the vertical and the horizontal Lagrange–Poincareoperator. The expressions that we obtain coincide with or can be easily derivedfrom the ones obtained in Cendra and Marsden (1987), Cendra, Ibort and Marsden(1987) and Marsden and Scheurle (1993b), with some changes in the notation. Westart with the covariant formulas for the vertical and horizontal Lagrange–Poincareoperators described in the previous theorems and the local expressions are theneasily derived.

Local Vertical Lagrange–Poincare Equation.

We now derive local coordinate expressions for the vertical Lagrange–Poincare equa-tions.

Suppose that Q has dimension n, so that Q/G has dimension r = n−dimG. Wechoose a local trivialization of the principal bundle Q→ Q/G to be X×G, where Xis an open set in Rr. Thus, we consider the trivial principal bundle π : X ×G→ Xwith structure group G acting only on the second factor by left multiplication.Let e be the neutral element of G and let A be a given principal connection onthe bundle Q → Q/G, or, in local representation, on the bundle X × G → X.We shall also assume that there are local coordinates xα, α = 1, . . . , r, in X andthat we choose the standard flat connection on X. Then, at any tangent vector(x, g, x, g) ∈ T(x,g) (X ×G) we have

A(x, g, x, g) = Adg (Ae(x) · x+ v)


where Ae is the g-valued 1-form on X defined by Ae(x) · x = A(x, e, x, 0) andv = g−1g. The vector bundle isomorphism αA in this case becomes

αA ([x, g, x, g]G) = (x, x)⊕ v

where v = (x,Ae(x) · x+ v). We will often write (x, x, v) instead of (x, x)⊕ v, andsometimes, simply v = Ae(x) · x+ v. Let us choose maps

eb : X → g,

where b = 1, ...,dim(G), such that, for each x ∈ X, is a basis of g. For each b =1, ..,dim(G), let eb(x) be the section of g given by eb(x) = [x, e, eb(x)]G ≡ (x, eb(x)).Let us call p = p(x, x, v) the vertical body momentum of the reduced system, thatis, by definition,

p(x, x, v) =∂l

∂v(x, x, v). (4.2.1)

We denote the components of p by pb = p(eb) ≡ 〈p, eb〉. We want to find an equationfor the evolution of pb. We have

d

dtpb =

d

dt〈p, eb〉

=⟨D

Dtp, eb

⟩+⟨p,D

Dteb

⟩. (4.2.2)

Using the vertical Lagrange-Poincare equation we obtain, immediately,⟨D

Dtp, eb

⟩= 〈p, [v, eb]〉

= 〈p, [Ae(x) · x+ v, eb]〉 . (4.2.3)

Lemma 2.3.4 gives the general formula for calculating the covariant derivative of agiven curve [q(t), ξ(t)]G in g,

D[q(t), ξ(t)]GDt

=[q(t),−[A (q(t), q(t)) , ξ(t)] + ξ(t)

]G. (4.2.4)

We are going to apply this formula for the case of the curve

eb (x(t)) = [x(t), e, eb (x(t))]G ≡ (x(t), eb (x(t)))

in g. Note that the tangent vector to the curve q(t) ≡ (x(t), e) is (q(t), q(t)) ≡(x(t), e, x(t), 0), and therefore A (q(t), q(t)) ≡ Ae (x(t)) · x. Using equation (4.2.4)we obtain

D

Dteb = [x, e,−[Ae(x) · x, eb] + eb]G

≡ (x,−[Ae(x) · x, eb] + eb) . (4.2.5)


Using equations (4.2.2), (4.2.3) and (4.2.5) we obtain the equation

dpbdt

= 〈p, [v, eb] + eb〉 . (4.2.6)

Using this equation we can easily find an expression in coordinates for the verticalLagrange-Poincare equation. Let us choose the functions eb(x) to be constant func-tions; therefore, we have eb = 0 and the equation for the evolution of pb becomes

dpbdt

= 〈p, [v, eb]〉 . (4.2.7)

Recall that v −Ae(x) · x = v, and thus the equation (4.2.7) becomes

dpbdt

= 〈p, [v −Ae(x) · x, eb]〉 . (4.2.8)

Let Cabd be the structure constants of the Lie algebra g. For the given local coor-dinates xα in X let Aaα(x) be the coefficients of Ae, that is, by definition, (Ae(x) ·x)aea = Aaα(x)xαea. Then equation (4.2.8) becomes the “Poincare part” of theLagrange-Poincare equations:

dpbdt

= pa(Cadbv

d − CadbAdαxα), (4.2.9)

This equation coincides with equation (5.3.3) of Bloch, Krishnaprasad, Marsden andMurray (1996) and equation (3.2) in Koon and Marsden (1997c). It also agrees, upto some sign problems, with that in Marsden and Scheurle (1993b) and is alsoimplicit in Cendra, Ibort and Marsden (1987).

Equation (4.2.9) reduces to the Euler–Poincare equation in the case that the baseis a point. If we think of the variables evolving as (x, x, v), and, thinking of pb as afunction of these variables, using the definition of pb, we can write this equation as

dpb(x, x, v)dt

= pa(x, x, v)(Cadbv

d − CadbAdα(x)xα). (4.2.10)

Notice that using the variable v, which is obtained by v = Ae(x) · x + v, we canwrite the equation as

dpb(x, x, Ae(x) · x+ v)dt

= pa(x, x, Ae(x)x+ v)Cadbvd. (4.2.11)

Observe that one can calculate pb in equation (4.2.1) by taking the derivative of lwith respect to either vb or vb.

In Bloch, Krishnaprasad, Marsden and Murray (1996), the variable vb is calledΩb and is interpreted as the locked body angular velocity. This variable is intrinsic,given the choice of a connection, whereas vb depends on the local trivialization. Infact, the form of equation (4.2.11) is dependent on choosing a local trivialization.


Local Horizontal Lagrange-Poincare Equation.

To calculate the local horizontal Lagrange-Poincare equation we shall first calculate

∂C l

∂x(x, x, v) · δx.

By Definition 3.3.3 of the notation ∂C l/∂x, we have

∂C l

∂x(x, x, v) · δx =

d

dλl (x+ λδx, x, w(λ))

∣∣∣∣λ=0

,

where w(λ) is a curve such that w(λ) ∈ gx+λδx for each λ, w(0) = v and

Dw(λ)Dλ

= 0.

If w(λ) = (x+ λδx, e, w(λ)), we can deduce from equation (4.2.4)

Dw(λ)Dλ

=(x(λ), e,− [A (x(λ)) · δx, w(λ)] +

dw(λ)dλ

). (4.2.12)

Therefore we must have

dw(λ)dλ

= [A (x(λ)) · δx, w(λ)] .

We then obtain

∂C l

∂x(x, x, v) · δx =

∂l

∂x(x, x, v) · δx+

∂l

∂v(x, x, v) · [A (x(λ)) · δx, w(λ)] .

On the other hand, it is easy to see that B(x)(x, δx) = (x, e,B(x, e)(x, δx)). Thenthe horizontal Lagrange-Poincare operator is(

∂l

∂x(x, x, v)− d

dt

∂l

∂x(x, x, v)

)δx =

∂l

∂v(x, x, v) (B(x, e)(x, δx) + [v, A(x) · δx]) .

(4.2.13)As we did with the vertical Lagrange-Poincare operator, it is convenient to rewritethis equation explicitly in coordinates and we easily obtain

∂l

∂xα(x, x, v)− d

dt

∂l

∂xα(x, x, v) =

∂l

∂va(x, x, v)

(Baβα(x, e)xβ + Cadbv

dAbα(x)),

(4.2.14)where a fixed basis ea of g has been chosen and, in this basis, v = vaea. Thisequation coincides with equation (5.3.2) of Bloch, Krishnaprasad, Marsden andMurray (1996) and equation (3.1) of Koon and Marsden (1997c). We remark thatin these papers the convention for the sign of the curvature Baαβ is the opposite tothe one used in this paper.


Summary.

The Lagrange-Poincare equations in coordinates have been shown to be (droppingthe independent variables from the notation)

dpbdt

= pa(Cadbv

d − CadbAdαxα)

(4.2.15)

∂l

∂xα− d

dt

∂l

∂xα=

∂l

∂va(Baβαx

β + CadbvdAbα

), (4.2.16)

where, as usual, a summation is implied over repeated indices.In the special case when the bundle Q→ Q/G is endowed with a trivial connec-

tion in local representation, i.e., A = 0, these equations simply become

dpbdt− paCadbvd = 0 (4.2.17)

∂l

∂xα− d

dt

∂l

∂xα= 0, (4.2.18)

which are the Hamel equations (Hamel (1904)).In many concrete applications, involving, for instance, stability theory, the general

form of the Lagrange-Poincare equations given in (4.2.15) and (4.2.16) are moreuseful than Hamel’s equations, as explained by an example in Marsden and Scheurle(1993b).

Coordinate Version of Wong’s Equations.

Locally, the expression of the Lagrangian l in the variables (x, x, v) is

l(x, x, v) =12κabv

avb +12gαβ(x)xαxβ .

The local expression of the vertical Lagrange-Poincare equation is given by (4.2.15),where

pa =∂l

∂va= κabv

b.

However, the first term on the right hand side of (4.2.15) equals

paCadbv

d = κaeveCadbv

d.

However, bi-invariance of κ means that

κaeCadb = κabC

aed

and so we get

paCadbv

d = κaeveCadbv

d = κabveCaedv

d = κab[v, v]a = 0.


Therefore, the vertical equation becomes

dpbdt

= −paCadbAdαxα.

The local expression of the horizontal Lagrange-Poincare equation is given by(4.2.16), where

∂l

∂xα=

12∂gβγ(x)∂xα

xβ xγ .

The second term on the right hand side of (4.2.16) vanishes as in the case of thevertical equation. Therefore the horizontal Lagrange-Poincare equation is given by

d

dt(gαβ(x)xβ) = −paBaβαxβ +

12∂gβγ(x)∂xα

xβ xγ ,

or equivalently, with pα := gαβ(x)xβ ,

d

dtpα = −paBaβαxβ −

12∂gβγ

∂xαpβpγ ,

which is the second Wong equation.

Page 53

5The Lie Algebra Structure on Sections ofthe Reduced Bundle

The main result of this section is the establishment of a natural Lie algebra structureon the space Γ(T (Q/G) ⊕ g) of sections of the bundle T (Q/G) ⊕ g, which will beused for reduction in the next section. This quotient Lie algebra structure is definedin Definition 5.2.3 and it is computed in Theorem 5.2.4. This Lie algebra is, roughlyspeaking, a synthesis of the Jacobi-Lie bracket of vector fields on shape space Q/Gwith the Lie algebra structure on the bundle g. However, as we shall see, this Liealgebra structure involves the reduced curvature form as well.

5.1 The Bundle T (Q/G)⊕ g Revisited

In this subsection, we assume that we have the following set up: a manifold Q, asmooth Lie group action ρ : G × Q → Q, and a connection A on the principalbundle π : Q→ Q/G.

Vertical and Horizontal Invariant Bundles.

Consider the vector bundle isomorphism αA : TQ/G → T (Q/G) ⊕ g defined inLemma 2.4.2 and let TQ = Hor(TQ) ⊕ Ver(TQ) be the decomposition into hori-zontal and vertical parts. Since the bundles Hor(TQ) and Ver(TQ) are G-invariantwe have

TQ/G = Hor(TQ)/G⊕Ver(TQ)/G.

This implies αA(Hor(TQ)/G) = T (Q/G) and αA(Ver(TQ)/G) = g.

54 5. The Lie Algebra Structure on Sections of the Reduced Bundle

Definition 5.1.1. Let

ιG(TQ) : IG(TQ)→ Q/G

be the vector bundle whose fiber IG(TQ)x = (ιG(TQ))−1(x) at an element x =[q]G ∈ Q/G is the vector space of all G-invariant vector fields along π−1(x). Thatis,

IG(TQ)x := Z : π−1(x)→ TQ | Z(q) ∈ TqQ for all q ∈ π−1(x) and g∗Z = Z.

Here and in what follows we denote by g∗ the pull back of various tensorial objectsby the diffeomorphism on Q defined by g ∈ G via the given G-action.

We also letιG(TQ)V : IVG (TQ)→ Q/G

be the vector bundle whose fiber IVG (TQ)x = (ιG(TQ)V )−1(x) at an element x =[q]G ∈ Q/G is the vector space of all vertical invariant vector fields on π−1(x). Thatis,

IVG (TQ)x := Y ∈ X∞(π−1(x)) | g∗Y = Y .

We call ιG(TQ)V : IVG (TQ)→ Q/G the vertical invariant bundle.Likewise, we define

ιG(TQ)H : IHG (TQ)→ Q/G

to be the vector bundle whose fiber IHG (TQ)x = (ιG(TQ)H)−1(x) at an elementx = [q]G ∈ Q/G is the vector space of all horizontal invariant vector fields on Qalong π−1(x). That is,

IHG (TQ)x := X : π−1(x)→ TQ | X(q) ∈ HorTqQ for all q ∈ Q and g∗X = X.

We call ιG(TQ)H : IHG (TQ)→ Q/G the horizontal invariant bundle.

Brackets of Invariant Vector Fields.

Recall that the Lie bracket of two G-invariant vector fields, say X and Y , is againa G-invariant vector field. Also, if X and Y are both vertical G-invariant vectorfields, that is, X,Y ∈ IVG (TQ), then [X,Y ] is also a vertical G-invariant vector field,that is, [X,Y ] ∈ IVG (TQ).

Given q ∈ Q, let x = [q]G, and define the diffeomorphism ρq : G → π−1(x)by ρq(g) = gq. This diffeomorphism ρq commutes with the action of G, that is,ρq(hg) = hρq(g), for all h, g ∈ G. We claim that IVG (TQ)x is isomorphic to the Liealgebra of left invariant vector fields on G. The Lie algebra of left invariant vectorfields X∞L (G) is identified with g = TeG in the usual way. That is, to each elementξ ∈ g, we associate the element Xξ ∈ X∞L (G) given by Xξ(g) = TeLg · ξ, where Lgdenotes the left translation map by the group element g ∈ G.

To prove the claim, we will show that the map given by the push-forward ofvector fields ρq∗ : X∞L (G) → IVG (TQ)x is a q-dependent Lie algebra isomorphism.

5.1 The Bundle T (Q/G)⊕ g Revisited 55

Indeed, this map preserves the Lie algebra structure since ρq commutes with theaction of G, and it is invertible, so it is a Lie algebra isomorphism.

Notice that the tangent map of ρq at e is given by Teρq · ξ = ξq. This implies inparticular that for any given ξ ∈ g, we have (ρq∗Xξ)(q) = ξq.

On the other hand, the map Tπ establishes a vector bundle isomorphism coveringthe identity on the base Q between IHG (TQ) and T (Q/G), namely, X ∈ IHG (TQ)x 7→Tqπ(X(q)) ∈ Tx(Q/G) for any q ∈ π−1(x) (it is easy to check that this definitiondoes not depend on the choice of q in the fiber over x).

The Bundles IVG (TQ) and g are Isomorphic.

The next lemma shows that there is a natural vector bundle isomorphism coveringthe identity between these two vector bundles that also preserves the Lie bracket.

Lemma 5.1.2. The mapβA : IVG (TQ)→ g

given byβA(Y ) = [q, A (Y (q))]G,

where Y ∈ IVG (TQ)x, x ∈ Q/G, and q ∈ π−1(x) is arbitrary, is a well defined Liealgebra bundle isomorphism. The inverse of βA is defined by the following condition:β−1A ([q, ξ]G) is the (unique) left invariant vector field Y on π−1(x) such that Y (q) =ξq.

Extend the domain of βA as follows: define the vector bundle isomorphism

βA : IHG (TQ)⊕ IVG (TQ)→ T (Q/G)⊕ g

in such a way that it coincides with the isomorphism given by Tπ on the summandIHG (TQ) and with the isomorphism βA defined above on the summand IVG (TQ).

Proof. To see that βA is well defined we simply check that for any g ∈ G we have

[gq,A (Y (gq))]G = [gq,A (gY (q))]G= [gq,Adg Y (q)]G= [q, A (Y (q))]G.

It is easy to check that βA is linear on each fiber. Now we will show that βAis a Lie algebra bundle isomorphism. Let us fix q ∈ Q. Then the pull-back ρ∗q :IVG (TQ)x → g is a Lie algebra isomorphism and we can easily check that, for allY ∈ IVG (TQ)x, and any q ∈ π−1(x) we have ρ∗q (Y ) = A (Y (q)). Therefore we have,for all X,Y ∈ IVG (TQ)x,

A ([X,Y ] (q)) = ρ∗q ([X,Y ])

= [ρ∗q(X), ρ∗q(Y )]

= [A (X(q)) , A (Y (q))] .


Using this and the definition of the Lie bracket on g (see Lemma 2.3.5) we can write

βA ([X,Y ]) = [q, A ([X,Y ] (q))]G= [q, [A (X(q)) , A (Y (q))]]G= [βA (X) , βA (Y )] .

It is clear that β−1A is given by the rule described in the statement since A (Y (q)) = ξ

if Y (q) = ξq. The rest of the proof, namely, to show that the extended βA is a vectorbundle isomorphism is obvious.

5.2 The Lie Algebra of Sections of T (Q/G)⊕ g

Quotient Vector Bundles.

We begin with some preliminaries concerning quotient vector bundles with someadditional structures.

Let τ : V → Q be a given vector bundle and let ρ : G × V → V denote a givenG-action on V (see §2.1).

Recall that, by definition, the action ρ is a vector bundle action if for eachg ∈ G the map ρg : V → V is a vector bundle isomorphism. This implies, inparticular, that there is an action ρ0 : G × Q → Q such that τ : V → Q isequivariant and for each q ∈ Q the restriction of ρg to τ−1(q) is a linear isomorphismρg : τ−1(q)→ τ−1(ρ0g(q)). We will often use the simpler notation gv and gq insteadof ρg(v) and ρ0g(q) respectively. As mentioned in §2.1, throughout this work we willassume that the action ρ0 of G on Q is free and, moreover, that relative to this actionQ→ Q/G is a principal G-bundle. Then V is also a principal G-bundle. Althoughthis assumption is not strictly needed for the validity of some properties, we willstill take it for granted, to simplify the exposition. An immediate consequence ofthis is that the quotient V/G carries a naturally defined vector bundle structureover the base Q/G.

More precisely, we have the following lemma, whose proof, which is standard, isincluded for the sake of completeness.

Lemma 5.2.1. The quotient V/G carries a naturally defined vector bundle struc-ture over the base Q/G, say τ/G : V/G → Q/G, where (τ/G)([v]G) is defined by(τ/G)([v]G) = [τv]G.

The projection πG(V ) : V → V/G is a surjective vector bundle homomorphismcovering π, and the restriction πG(V )|τ−1(q) : τ−1(q)→ (τ/G)−1([q]G) is a linearisomorphism for each q ∈ Q. In addition, πG(V )|τ−1(q) g−1 = πG(V )|τ−1(gq) forall q ∈ Q and g ∈ G.

Assume that τ ′ : V ′ → Q′ is another vector bundle and that there is a vectorbundle action G× V ′ → V ′. Let f : V → V ′ be an equivariant vector bundle map.Then the naturally induced quotient map [f ]G : V/G → V ′/G is a vector bundlemap. If f is a vector bundle isomorphism, so is [f ]G.

5.2 The Lie Algebra of Sections of T (Q/G)⊕ g 57

Proof. First we show that τ/G is well defined, namely, for any g ∈ G and anyv ∈ V we have τ/G([gv]G) = [τgv]G = [gτv]G = [τv]G. Next we define the vectorspace structure on each fiber (τ/G)−1([q]G). Let [vi]G ∈ V/G, for i = 1, 2, be suchthat τ/G[v1]G = τ/G[v2]G = [q]G, for some q ∈ Q. This implies that there aregi ∈ G such that giτvi = q for i = 1, 2. The gi are uniquely determined for a fixedq, as before, because the action ρ0 is free. Define [v1]G + [v2]G = [g1v1 + g2v2]G.We must show that this gives a well defined additive structure. Elements of Vequivalent to vi are of the type hivi with hi ∈ G for i = 1, 2. For given hivi withhi ∈ G, i = 1, 2, the only elements di ∈ G such that diτhivi = q are di = gih

−1i ,

i = 1, 2. Then our definition gives

[h1v1]G + [h2v2]G = [d1h1v1 + d2h2v2]G = [g1v1 + g2v2]G= [v1]G + [v2]G.

This shows that the definition does not depend on the choice of the representativein the class [vi]G, i = 1, 2, for a given q. Now, if we choose an arbitrary element,say hq ∈ [q]G, then the only elements di ∈ G such that diτvi = hq are di = hgi, fori = 1, 2. We obtain then

[v1]G + [v2]G = [hg1v1 + hg2v2]G= [h(g1v1 + g2v2)]G= [g1v1 + g2v2]G

as before.We can define λ[v]G = [λv]G and check that it is well defined in a similar way.

To finish the proof, it only remains to show that the restriction of πG(V ) to eachfiber, πG(V ) : τ−1(q)→ (τ/G)−1([q]G), is a linear isomorphism. This can be easilyestablished using the definition of the linear structure on (τ/G)−1([q]G). We omitthe rest of the proof, which can also be easily performed using standard techniques.

Spaces of Sections.

For a vector bundle τ : V → Q, the vector space of sections of V is denoted by Γ(V ),which is also a C∞(Q)-module. Let f : V1 → V2 be a vector bundle homomorphismwhere τ i : Vi → Qi are vector bundles for i = 1, 2. This implies, in particular,that there is a map f0 : Q1 → Q2 such that, for each q ∈ Q1, the restrictionf : (τ1)−1(q) → (τ2)−1 (f0(q)) is linear. Assume, in addition, that this restrictionis a linear isomorphism. Then f induces a linear map f∗ : Γ(V2) → Γ(V1) definedby f∗(s)(q) = f−1 (s (f0(q))), where f−1 is the inverse of the restriction of f to(τ2)−1(q). The properties (f h)∗ = h∗ f∗ and id∗V = idΓ(V ), where both f andh satisfy the condition that their restriction to each fiber is a linear isomorphism,can be easily checked. It follows, in particular, that if f is an isomorphism, thenf−1∗ = f∗−1. In this case we write f−1∗ = f∗.

Sections si ∈ Γ(Vi), i = 1, 2, are said to be f-related if for all q ∈ Q1 we havef (s1(q)) = s2 (f0(q)). We can easily show that if for each q ∈ Q1 the restriction


f : (τ1)−1(q)→ (τ2)−1 (f0(q)) is a linear isomorphism as before, then for any givens2 ∈ Γ(V2), the section s1 = f∗(s2) is the only one in Γ(V1) which is f -related tos2.

If G × V → V is a vector bundle action on τ : V → Q, a section s : Q → V iscalled an invariant section if for all g ∈ G and all q ∈ Q we have gs(q) = s(gq).The set ΓG(V ) of invariant sections of V is a subspace of Γ(V ).

Lemma 5.2.2. Let G × V → V be a vector bundle action and let πG(V ) : V →V/G be the vector bundle homomorphism described in lemma 5.2.1. Then (πG(V ))∗ :Γ(V/G)→ ΓG(V ) is a linear isomorphism.

Proof. Let s ∈ Γ(V/G). Then for each q ∈ Q and each g ∈ G, using the fact thatthe restriction of πG(V ) to each fiber is an isomorphism, we have

πG(V )∗s(gq) = πG(V )−1 (s([gq]G))

= g(πG(V )−1 (s([q]G))

)= gπG(V )∗s(q),

where in the first equation πG(V ) is restricted to τ−1(gq) and in the second πG(V )is restricted to τ−1(q). The second equality is then an easy consequence of thedefinition of a vector bundle action and, also, the definition of the quotient vectorbundle (see Lemma 5.2.1). Thus, we have shown that πG(V )∗ is an injective mapinto ΓG(V ). Now let s ∈ ΓG(V ). Define s ∈ Γ(V/G) by s([q]G) = [s(q)]G. Theelement s is well defined because, since s is invariant, for any g ∈ G and any q ∈ Qwe have [s(gq)]G = [gs(q)]G = [s(q)]G. We can easily check that s = πG(V )∗s whichfinishes the proof.

Quotient Lie Algebras.

If v ∈ ΓG(V ) we will denote [v]G or, sometimes, v the corresponding section ofΓ(V/G) via the isomorphism πG(V )∗ of the previous lemma. Let ρ be a vectorbundle action of the Lie group G on the vector bundle τ : V → Q. Then we obtaina representation ρ∗ : G×Γ(V )→ Γ(V ) given by the operation (g, s) ∈ G×Γ(V ) 7→ρg∗s ∈ Γ(V ). We will often write simply g∗s instead of ρg∗s. It is clear that ΓG(V )is an invariant subspace of Γ(V ). Now assume that there is a Lie algebra structureon Γ(V ) which is invariant under the action ρ∗, that is, g∗[s1, s2] = [g∗s1, g∗s2] forall si ∈ Γ(V ), i = 1, 2, and all g ∈ G. In particular, ΓG(V ) is a Lie subalgebra ofΓ(V ).

Since πG(V )∗ : Γ(V/G) → ΓG(V ) is a linear isomorphism we can define a Liealgebra structure on Γ(V/G) in the following way.

Definition 5.2.3. Assume that the space of sections Γ(V ) of the vector bundleτ : V → Q has a Lie algebra structure. The quotient Lie algebra structure onΓ(V/G) is defined by

[s1, s2] = (πG(V )∗)−1 [πG(V )∗s1, πG(V )∗s2] .


The most important case of the situation described above is the case of the vectorbundle TQ on which G acts by the tangent lift of the action of G on Q and the Liebracket on Γ(TQ) ≡ X∞(Q) is the usual Lie bracket of vector fields. According tothe previous results we obtain a quotient Lie algebra structure on Γ(TQ/G).

Let us denote π ≡ πG(Q) the natural projection of the principal bundle π : Q→Q/G, for simplicity. Recall that for each choice of a principal connection A on Qwe have vector bundle isomorphisms

αA : TQ/G→ T (Q/G)⊕ g

andβA : IHG (TQ)⊕ IVG (TQ)→ T (Q/G)⊕ g.

The linear isomorphism α∗A between the corresponding spaces of sections definesa Lie algebra structure on Γ (T (Q/G)⊕ g) by declaring it to be a Lie algebraisomorphism. In order to calculate this Lie bracket on Γ (T (Q/G)⊕ g) we shallmake use of the equivalent condition that β∗A is a Lie algebra isomorphism. We dothis in the next theorem.

The map Tπ induces a well defined isomorphism between Γ(IHG (TQ)

)and X∞(Q/G).

More precisely, for any X ∈ Γ(IHG (TQ)

)the vector field π∗X ∈ X∞(Q/G) given

by π∗X(x) = TπX(q), where q ∈ π−1(x) is arbitrary, is well defined. Also, ifX,Y ∈ Γ

(IHG (TQ)

)then π∗[X,Y ] = [π∗X,π∗Y ] since the bracket operation pre-

serves π-relatedness of vector fields. Observe that the inverse of π∗ : Γ(IHG (TQ)

)→

X∞(Q/G) is given by the horizontal lift of vector fields, h : X∞(Q/G)→ Γ(IHG (TQ)

),

which also coincides with the restriction of β∗A to X∞(Q/G). Also, we recall thatif X,Y ∈ Γ

(IHG (TQ)

)then A([X,Y ]) = −B(X,Y ) where B is the curvature of A.

Finally we remark that, from what we have said before, we can deduce that thereare natural identifications

ΓG(TQ) ≡ Γ(TQ/G) ≡ Γ(IHG (TQ)

)⊕ Γ

(IVG (TQ)

),

where the last identification involves the choice of a connection.

Calculation of the Lie Algebra Structure.

The main result of this section is a formula, given in the next theorem, for the Liebracket on the Lie algebra Γ (T (Q/G)⊕ g) ≡ X∞(Q/G)⊕ Γ(g) which involves theLie bracket on g, the connection ∇A on g, and the g-valued curvature BA.

Theorem 5.2.4. Let Xi ⊕ ξi ∈ Γ (T (Q/G)⊕ g), i = 1, 2, be given sections ofT (Q/G)⊕ g. Then

[X1 ⊕ ξ1, X2 ⊕ ξ2] = [X1, X2]⊕ ∇AX1ξ2 − ∇AX2

ξ1 − BA(X1, X2) + [ξ1, ξ2].

A remark is in order to avoid any confusion in the interpretation of the bracketnotation in the preceding and in several other formulas. Namely, the bracket on theleft-hand-side is the bracket in Γ (T (Q/G)⊕ g) given in Definition 5.2.3, while the


bracket that appears immediately before the sign ⊕ is the usual bracket of vectorfields. This caution is needed to avoid the apparently contradictory statement

[X1, X2] ≡ [X1 ⊕ 0, X2 ⊕ 0] = [X1, X2]⊕−BA(X1, X2).

Proof of the Theorem.. Let Xi⊕ ξi ∈ Γ (T (Q/G)⊕ g), i = 1, 2. Then there areelements Yi ∈ Γ

(IVG (TQ)

)and Xh

i ∈ Γ(IHG (TQ)

), where Xh

i is the horizontal liftof Xi ∈ X∞(Q/G), such that β∗Aξi = Yi and β∗AXi = Xh

i for i = 1, 2. According tothe definition of the bracket in Γ (T (Q/G)⊕ g) given before the theorem, we have

[X1 ⊕ ξ1, X2 ⊕ ξ2] = βA∗[Xh1 + Y1, X

h2 + Y2].

For any q ∈ Q let x = π(q) = [q]G. By (2.2.4) we have

[Xh1 , X

h2 ](q) = [X1, X2]h(q)−B

(Xh

1 (q), Xh2 (q)

)q.

Applying βA∗ to this identity and evaluating the result at x = [q]G, we obtain, inthe Lie algebra Γ (T (Q/G)⊕ g),

[X1, X2](x) = [X1, X2](x)⊕[q,−B

(Xh

1 (q), Xh2 (q)

)]G

= [X1, X2](x)⊕−BA(x)(X1, X2);

again, the bracket [X1, X2] on the left hand side is that in Γ (T (Q/G)⊕ g), whereasthe same notation on the right hand side signifies the usual Lie bracket of vectorfields. Using Lemma 5.1.2 we can deduce that for any x ∈ Q/G,

βA∗[Y1, Y2](x) = [ξ1, ξ2](x).

Thus, we are left with the computation of the terms [X1, ξ2] = βA∗[Xh1 , Y2] and

[X2, ξ1] = βA∗[Xh2 , Y1]. For this, we first recall that, since Xh

1 is an invariant hori-zontal vector field, its flow Xh

1t, which is the horizontal lift of the flow X1t of X1,maps a point q ∈ π−1(x) to a point Xh

1t(q) ∈ π−1 (X1t(x)). Since Xh1 and X1 are

π-related, we have π Xh1t = X1t π, and thus, for any vertical vector field Y on

Q, the pull back Xh∗1t Y is also vertical. Therefore the Lie bracket

[Xh1 , Y ] =

d

dtXh∗

1t Y

∣∣∣∣t=0

is also a vertical vector field. Using this, we see that for any q ∈ Q we have

[Xh1 , Y2](q) = A

([Xh

1 , Y2](q))q.

The G-invariance of Xh1 and Y2, immediately implies the G-invariance of [Xh

1 , Y2].Thus [Xh

1 , Y2] is a vertical G-invariant vector field.Second, we need to calculate βA∗[Xh

1 , Y2]. For any x ∈ Q/G and any q ∈ π−1(x),we have

βA∗[Xh1 , Y2](x) =

[q, A

([Xh

1 , Y2](q))]G.


Cartan’s structure equations give

dA(Xh1 , Y2) =

[A(Xh

1 ), A(Y2)]

+B(Xh1 , Y2).

Since A(Xh1 ) = 0 (because Xh

1 is horizontal) and B(Xh1 , Y2) = 0 (because Y2 is

vertical), we obtain dA(Xh1 , Y2) = 0. On the other hand we have the formula

dA(Xh1 , Y2) = Xh

1 [A(Y2)]− Y2

[A(Xh

1 )]−A

([Xh

1 , Y2]).

We conclude that A([Xh

1 , Y2])

= Xh1 [A(Y2)]. This shows, in particular, that at a

given point q ∈ Q, A([Xh

1 , Y2](q))

only depends on Xh1 (q) and not on the behavior

of Xh1 in a neighborhood of q. Now it is clear that, for any q ∈ Q, we have

(Xh

1 [A(Y2)])

(q) =d

dtA(Y2

(Xh

1t(q)))∣∣∣∣

t=0

.

Let q ∈ Q and x = π(q); thusX1t(x) = π(Xh

1t(q)), for all t. Let ξ2(t) = A

(Y2

(Xh

1t(q)))

for all t. Lemma 5.1.2 implies that ξ2 (x(t)) =[Xh

1t(q), ξ2(t)]G

for all t and therefore

∇AX1(x)ξ2(x) =D

Dt

[Xh

1t(q), ξ2(t)]G

∣∣∣∣t=0

.

Now using Lemma 2.3.4 to calculate the covariant derivative of the curve [Xh1t(q), ξ(t)]G ∈

g we obtainD

Dt

[Xh

1t(q), ξ2(t)]G

∣∣t=0

= [q, ξ2(0)]G.

But ξ2(0) =(Xh

1A(Y2))

(q) which implies ξ2(0) = A([Xh

1 , Y2](q)). We conclude

that∇AX1

ξ2 = βA∗[Xh1 , Y2].

Analogously, we have ∇AX2ξ1 = βA∗[Xh

2 , Y1].Collecting these results, we obtain

[X1 ⊕ ξ1, X2 ⊕ ξ2] = [X1, X2]⊕ ∇AX1ξ2 − ∇AX2

ξ1 − B(X1, X2) + [ξ1, ξ2],

as desired.

The following corollary is a consequence of Theorem 5.2.4.

Corollary 5.2.5. Let pQ/G : T (Q/G) ⊕ g → T (Q/G) and pg : T (Q/G) ⊕ g → gbe the natural projections. Then the induced maps p(Q/G)∗ : Γ (T (Q/G)⊕ g) →Γ (T (Q/G)) given by p(Q/G)∗(X ⊕ ξ) = X and p∗g : Γ (g) → Γ (T (Q/G)⊕ g) givenby p∗g(ξ) = 0⊕ ξ are Lie algebra homomorphisms. The Lie algebra structure on Γ (g)is defined pointwise, that is, for given ξ, η ∈ Γ (g) we have, [ξ, η](x) = [ξ(x), η(x)]for all x ∈ Q/G.


Page 63

6Reduced Tangent Bundles

The results of the preceding sections may be viewed as a geometrized and intrinsicway of writing the results of Cendra, Ibort and Marsden (1987) and of Marsden andScheurle (1993b). Next we turn to our main task: find a context in which the La-grangian reduction process can be iterated . In other words, we develop a frameworkin which the objects are stable under Lagrangian reduction. These objects will becalled reduced tangent bundles.

Lagrange–Poincare Bundles and Reduced Tangent Bundles.

We begin by describing the geometric objects on which reduced Lagrangians arenaturally defined. They form a category of vector bundles denoted LP, objectsof which will be called Lagrange–Poincare bundles. Important special Lagrange–Poincare bundles are the reduced tangent bundles. These form a subcategory, de-noted RT, which is the smallest subcategory that contains the tangent bundles andwhich remains stable under reduction; each bundle carries some additional struc-ture needed to describe the reduction of given Lagrangians defined on them andthe corresponding variational principles.

For example, objects of RT are vector bundles of the type T (Q/G)⊕ g where, aswe have seen, g is a Lie algebra bundle, there is a covariant derivative ∇A on g, ag-valued 2-form B, and the space of sections of T (Q/G) ⊕ g admits a Lie algebrastructure which is related to ∇A, B, and the Lie algebra structure on the fibers ofg by the formula of Theorem 5.2.4.

In this section we show how to reduce further these kinds of objects. They are,in a sense, special cases of Lie algebroids (see Mackenzie (1987), Courant (1990),Weinstein [1996, 1998] and Cannas da Silva and Weinstein (1999)) although we

64 6. Reduced Tangent Bundles

consider some extra structure on them. We also recall (see Weinstein (1996) andMarsden et al. (2000)) that these notions are very useful in discrete Lagrangianmechanics. Further exploration of the link between our work, that of groupoids andalgebroids, as well as noncommutative differential geometry would of course be veryinteresting.

Below we will show that the reduction process can be performed by stages andwe will also write explicit expressions for Lagrangians and variational principlesreduced by stages.

Even though the most important objects of LP seem to be those of RT, it isnatural to first deal with the larger category LP rather than the category RT.

The objects in the category LP of Lagrange–Poincare bundles are vector bundlesof the type TQ ⊕ V where V is a Lie algebra bundle over the base Q and whichcarry

1. a covariant derivative D on V ,

2. a V -valued 2-form on Q, and

3. a Lie algebra structure on Γ(TQ⊕V ) that satisfies an equality formally similarto the formula of Theorem 5.2.4.

We shall make this more explicit and describe the mappings in this category shortly.We will show that the category RT is strictly contained in the category LP. More-

over, the simplest way of describing RT is by defining it as the smallest subcategoryof LP that contains the tangent bundles TQ. The special importance of the cate-gory RT is that each of its objects is a reduced tangent bundle, after some numberof reduction by stages; in addition, reduced versions of Euler–Lagrange equationscorresponding to invariant Lagrangians on tangent bundles can be written in termsof these reduced tangent bundles.

It is important to remark at this point that we can obtain a generalization ofall this, that is, a category bigger than LP, which contains in particular vectorbundles which are some subbundles of bundles of LP, to describe reduction in non-holonomic mechanics. This, as well as other interesting topics like a local study ofobjects of LP, will be the purpose of future work.

An important result in this section is an explicit expression for the reductionby stages of the Lie algebra of sections of bundles which are objects of LP. This,together with the results of §5.2, will be related in §8 to the Poisson bracket onthe dual bundles of the bundles which are objects of LP. These dual bundles carrya Poisson bracket induced by the structure of the objects of LP. Therefore, byduality, we obtain a direct link to the topic of Poisson reduction by stages of atleast some important examples of Poisson manifolds. The special case of duals ofelements of RT gives cotangent Poisson reduction (see Montgomery, Marsden andRatiu (1984) and Montgomery (1986)). A generalization of all this for more generalPoisson manifolds is tied to the consideration of a category bigger than LP asindicated above, and will be also the purpose of future work.

6.1 The Geometry of Lagrange–Poincare Bundles 65

6.1 The Geometry of Lagrange–Poincare Bundles

We will need to fix some notation for maps induced on quotients. Recall that if aLie group G acts on the manifolds E and F and f : E → F is a G-equivariant map,then there is a natural quotient map f/G : E/G→ F/G, defined by

f/G ([a]G) = [f(a)]G .

Lagrange–Poincare Bundles.

We now give the details of the definition of a Lagrange–Poincare bundle.

Definition 6.1.1. The category LP of Lagrange–Poincare bundles is definedas follows:

(A) The objects of LP are vector bundles which are Whitney sums of the formτQ⊕τ : TQ⊕V → Q where τQ : TQ→ Q is the tangent bundle of a manifoldQ and τ : V → Q is a vector bundle, together with some extra structure givenby:

(a) a Lie algebra structure on each fiber of V , in such a way that V is aLie algebra bundle; the Lie bracket of given elements v1, v2 ∈ τ−1(q) isdenoted by [v1, v2];

(b) a V -valued 2-form ω on Q;

(c) a covariant derivative D/Dt for curves in V , related in the standard wayto a connection ∇ on V , namely, if v(t) is any curve in V , consider thecurve q(t) = τ (v(t)) on Q, and let

T t+st : τ−1q(t+ s)→ τ−1q(t)

be the parallel transport along q(t) defined by ∇; then

Dv(t)Dt

=d

dsT t+st v(t+ s)

∣∣∣∣s=0

;

(d) a Lie bracket operation defined on sections X ⊕u ∈ Γ(TQ⊕V ) which isdefined by

[X1 ⊕ u1, X2 ⊕ u2] = [X1, X2]⊕∇X1u2 −∇X2u1 − ω(X1, X2) + [u1, u2].

(B) Let TQi ⊕ Vi, i = 1, 2, be two Lagrange-Poincare bundles with structures[ , ]i, ωi, ∇i on τ i : Vi → Qi, i = 1, 2. Let Di/Dit denote the covariantderivative along a curve in Qi defined by ∇i on Vi, i = 1, 2. A morphismf : TQ1 ⊕ V1 → TQ2 ⊕ V2 between Lagrange–Poincare bundles is a vectorbundle map covering f0 : Q1 → Q2 that satisfies the following conditions:

(a) f(TQ1) ⊂ TQ2 and, moreover, f |TQ1 = Tf0;


(b) f(V1) ⊂ V2 and the restricted vector bundle map f |V1 commutes withthe structures on Vi given by [ , ]i, ωi, ∇i, which means that for givenu, u′ ∈ (τ1)−1(q), X,X ′ ∈ τ−1

Q1(q), and a given curve v(t) in V1, the

following conditions are satisfied

f ([u, u′]1) = [f(u), f(u′)]2,

f (ω1(X,X ′)) = ω2 (f(X), f(X ′)) ,

and

f

(D1v(t)D1t

)=D2f (v(t))

D2t.

Bundles of the type T (Q/G)⊕ g as considered in previous sections are importantexamples of elements of LP.

Projections and Injections.

The following lemma is an easy consequence of the previous definition.

Lemma 6.1.2. Let TQ⊕ V be an object of LP and let ∇, ω, [ , ] be the structureon V . Then the following statements hold.

(i) The maps Γ(TQ⊕V )→ Γ(TQ) given by X⊕v 7→ X and Γ(V )→ Γ(TQ⊕V )given by v 7→ 0⊕ v are Lie algebra homomorphisms.

(ii) Let ϕ ∈ C∞(Q), Xi ∈ Γ(TQ), and vi ∈ Γ(V ), i = 1, 2, be given. Then wehave

[v1, ϕv2] = ϕ[v1, v2][X1, ϕX2] = X1[ϕ]X2 + ϕ[X1, X2][X1, ϕv2] = X1[ϕ]v2 + ϕ[X1, v2][ϕX1, v2] = ϕ[X1, v2]

[X1 ⊕ v1, ϕ(X2 ⊕ v2)] = X1[ϕ](X2 ⊕ v2) + ϕ[X1 ⊕ v1, X2 ⊕ v2].

Let us denote W := TQ ⊕ V and also wi := Xi ⊕ vi, for i = 1, 2. Define w1[ϕ] :=X1[ϕ]. Then the previous equalities can be converted into a single equality

[w1, ϕw2] = ϕ[w1, w2] + w1[ϕ]w2.

Morphisms between Lagrange–Poincare Bundles.

The nature of morphisms between Lagrange–Poincare bundles is clarified by thefollowing.

Lemma 6.1.3. (i) Let TQ ⊕ V be an object of LP and let ∇, ω, [ , ] be thestructure on V . Assume that there is a structure ∇′, ω′, [ , ]′ on V such thatthe Lie algebra on Γ(TQ ⊕ V ) defined by ∇′, ω′, [ , ]′ is the same as the Liealgebra defined by ∇, ω, [ , ]. Then ∇′ = ∇, ω′ = ω, and [ , ]′ = [ , ].


(ii) Let TQi ⊕ Vi be objects of LP for i = 1, 2. Let f : TQ1 ⊕ V2 → TQ2 ⊕ V2 bea vector bundle isomorphism. Assume that

f∗ : Γ(TQ1 ⊕ V2)→ Γ(TQ2 ⊕ V2)

is a Lie algebra homomorphism. Then f(V1) = V2. Assume, in addition, thatf(TQ1) ⊂ TQ2. Then f is an isomorphism in the category LP, that is, f isan isomorphism of Lagrange–Poincare bundles.

Proof. (i) By hypothesis, we have, for all Xi ⊕ ui ∈ Γ(TQ⊕ V ), i = 1, 2,

[X1, X2]⊕∇X1u2 −∇X2u1 − ω(X1, X2) + [u1, u2]= [X1, X2]⊕∇′X1

u2 −∇′X2u1 − ω′(X1, X2) + [u1, u2]′.

Taking u1 = u2 = 0 and Xi arbitrary for i = 1, 2, we obtain ω′ = ω. TakingX1 = X2 = 0 and ui arbitrary for i = 1, 2, we obtain [ , ]′ = [ , ]. Taking u1 = 0,X2 = 0 and u2, X1 arbitrary, we obtain ∇′ = ∇.

(ii) First, we will prove that f(V1) = V2. By contradiction, assume that thereis an element v1 ∈ V1 such that f(v1) = X2 ⊕ v2, where X2 ∈ Tq2Q2, satisfiesX2 6= 0. Then there exists ϕ2 ∈ C∞(Q2) such that dϕ(q2)(X2) 6= 0. Since f isa vector bundle isomorphism, the induced map on the base f0 : Q1 → Q2 is adiffeomorphism. Let q1 = f−1

0 (q2) and ϕ1 = f∗0ϕ2 = ϕ2 f0. Let v1 ∈ Γ(V1) be suchthat v1(q1) = v1 and let f∗v1 = X2 ⊕ v2. Thus, X2(q2) ⊕ v2(q2) = X2 ⊕ v2. Weobviously have [ϕ1v1, v1] = ϕ1[v1, v1] = 0, and therefore f∗[ϕ1v1, v1] = 0.

On the other hand, using the fact that f∗ is a Lie algebra isomorphism and alsoLemma 6.1.2, we have

f∗[ϕ1v1, v1] = [ϕ2(X2 ⊕ v2), X2 ⊕ v2]= −X2[ϕ2](X2 ⊕ v2) + ϕ2[X2 ⊕ v2, X2 ⊕ v2]= −X2[ϕ2](X2 ⊕ v2),

which gives a contradiction since X2[ϕ2](q2) = dϕ(q2)(X2) 6= 0. We have provenhence that f(V1) ⊂ V2.

Replacing in the above argument f by the vector bundle isomorphism f−1 :TQ2 ⊕ V2 → TQ1 ⊕ V1 and using the fact that (f−1)∗ is a Lie algebra isomor-phism between the corresponding spaces of sections, it follows that f−1(V2) ⊂ V1.Therefore, f(V1) = V2.

Now let ∇i, ωi, [ , ]i be the structure on the bundle Vi for i = 1, 2, and assumethat f(TQ1) ⊂ TQ2. Since f is a vector bundle isomorphism and f(V1) = V2, itfollows that f(TQ1) = TQ2. However f∗ is a Lie algebra isomorphism so we get for


any X ⊕ v,X ′ ⊕ v′ ∈ Γ(TQ1 ⊕ V1),

f∗[X,X ′]⊕ f∗∇1Xv′ − f∗∇1X′v − f∗ω1(X,X ′) + f∗[v, v′]1

= f∗ ([X,X ′]⊕∇1Xv′ −∇1X′v − ω1(X,X ′) + [v, v′]1)

= f∗[X ⊕ v,X ′ ⊕ v′]1= [f∗(X ⊕ v), f∗(X ′ ⊕ v′)]2= [f∗X ⊕ f∗v, f∗X ′ ⊕ f∗v′]2= [f∗X, f∗X ′]⊕∇2f∗Xf∗v

′ −∇2f∗X′f∗v − ω2(f∗X, f∗X ′) + [f∗v, f∗v′]2.

Taking X = X ′ = 0, we get f∗[v, v′]1 = [f∗v, f∗v′]2. Next, taking X ′ = 0 andv = 0 we obtain f∗∇1Xv

′ = ∇2f∗Xf∗v′. Finally, taking v = v′ = 0, we have

f∗ω1(X,X ′) = ω2(f∗X, f∗X ′) and also f∗[X,X ′] = [f∗X, f∗X ′].To show that f is a morphism of Lagrange–Poincare bundles (i.e., a morphism

in the category LP), it remains to show that f |TQ1 = Tf0. Let X ∈ Γ(TQ1) =X∞(Q1) and ϕ ∈ C∞(Q1). Then we have

f∗[X,ϕX] = f∗ (X[ϕ]X) = f∗ (X[ϕ]) f∗X.

On the other hand,

f∗[X,ϕX] = [f∗X, f∗(ϕX)] = [f∗X, f0∗ϕf∗X] = ((f∗X)[f0∗ϕ]) (f∗X).

Therefore we have proven that for any X ∈ Γ(TQ1) and ϕ ∈ C∞(Q1) we havef∗ (X[ϕ]) = (f∗X)[f0∗ϕ]. Since f(TQ1) = TQ2, this implies f |TQ1 = Tf0 as weshall show below.

Let q2 ∈ Q2 and denote q1 = f−10 (q2). Then using the above relation in the third

equality below, we get

d(ϕ f−10 )(q2) · Tq1f0(X(q1)) = dϕ(q1) ·X(q1) =

(X[ϕ] f−1

)(q2)

= (f∗X)[f0∗ϕ](q2) = d(ϕ f−10 )(q2) · (f∗X)(q2)

= d(ϕ f−10 )(q2) · f (X(q1)) ,

that is, f (X(q1)) = Tq1f0(X(q1)) for any X1 ∈ X∞(Q1) which proves that f |TQ1 =Tf0.

Morphism-related Sections of Lagrange–Poincare Bundles.

Let f : TQ1 ⊕ V1 → TQ2 ⊕ V2 be a given morphism of Lagrange–Poincare bundlesand let X ⊕ u ∈ Γ(TQ1 ⊕ V1) and Y ⊕ v ∈ Γ(TQ2 ⊕ V2) be given. Then X ⊕ u andY ⊕ v are said to be f-related if

f (X(q)⊕ u(q)) = Y (f0(q))⊕ v (f0(q))

for all q ∈ Q1. In particular, it is easy to prove that if X⊕u and Y ⊕v are f -relatedthen X and Y are f0-related as vector fields according to the usual definition.


Lemma 6.1.4. Let f : TQ1⊕V1 → TQ2⊕V2 be a morphism of Lagrange–Poincarebundles and let Xi⊕ui ∈ Γ(TQ1⊕V1), Yi⊕vi ∈ Γ(TQ2⊕V2) for i = 1, 2, be given.Assume that Xi⊕ ui and Yi⊕ vi are f -related for i = 1, 2. Then [X1⊕ u1, X2⊕ u2]and [Y1 ⊕ v1, Y2 ⊕ v2] are also f -related.

Proof. Since Xi⊕ui and Yi⊕vi are f -related for i = 1, 2, Definition 6.1.1 impliesfor all q ∈ Q1

f ([X1 ⊕ u1, X2 ⊕ u2](q))= f (([X1, X2]⊕∇X1u2 −∇X2u1 − ω(X1, X2) + [u1, u2]) (q))= [Y1, Y2](f0(q))⊕∇Y1v2 −∇Y2v1 − ω(Y1, Y2) + [v1, v2] (f0(q))= [Y1 ⊕ v1, Y2 ⊕ v2](f0(q)).

Lemma 6.1.5. Let f : TQ1⊕V1 → TQ2⊕V2 be a morphism in LP and assume,in addition, that f is a linear isomorphism on each fiber. Then f∗ : Γ(TQ2⊕V2)→Γ(TQ1 ⊕ V1) is a Lie algebra homomorphism.

Proof. Using the remarks preceding Lemma 5.2.2 it follows that f∗ is well defined,that for given Yi⊕ vi ∈ Γ(TQ2⊕ V2), i = 1, 2, f∗(Yi⊕ vi) ∈ Γ(TQ1⊕ V1), and thatf∗(Yi ⊕ vi) = f∗Yi ⊕ f∗vi and Yi ⊕ vi are f -related for i = 1, 2. Then, by Lemma6.1.4, [f∗(Y1⊕v1), f∗(Y2⊕v2)] and [Y1⊕v1, Y2⊕v2] are also f -related. However, thehypothesis of the lemma implies that there is one and only one section of TQ1⊕V1

which is f -related to a given section of TQ2 ⊕ V2, so we can conclude

[f∗(Y1 ⊕ v1), f∗(Y2 ⊕ v2)] = f∗ ([Y1 ⊕ v1, Y2 ⊕ v2]) .

Group Actions on Vector Bundles with a Structure [ , ], ω, ∇.

As we have already indicated, when tangent bundles are reduced, one enters thecategory of Lagrange–Poincare bundles and so, to continue the process of reduction,it is important to know how these objects themselves behave under reduction. Thefirst job is to examine group actions on vector bundles with the extra structuregiven in Definition 6.1.1.

Let τ i : Vi → Qi, i = 1, 2, be vector bundles with extra structure [ , ]i, ωi,∇i satisfying conditions (a), (b), (c) of Definition 6.1.1A. A vector bundle mapf : V1 → V2 is called a morphism if it commutes with the structures given by [ , ]i,ωi, ∇i, i = 1, 2, that is, if the conditions of Definition 6.1.1B(b) hold.

Definition 6.1.6. An action ρ : G×V → V of a Lie group G on a vector bundleτ : V → Q with extra structure [ , ], ω, ∇ as in Definition 6.1.1A , is a vector bundleaction such that, for each g ∈ G, ρg : V → V is a morphism in the previous sense.More precisely, we assume


(a) g[v1, v2] = [gv1, gv2] for all g ∈ G and all v1, v2 ∈ V satisfying τ(v1) = τ(v2);

(b) gω(X,Y ) = ω(gX, gY ), for all g ∈ G and all X,Y ∈ TQ satisfying τQ(X) =τQ(Y );

(c)Dgv(t)Dt

= gDv(t)Dt

for any curve v(t) on V and any g ∈ G.

Now we shall state the following lemma whose proof is straightforward.

Lemma 6.1.7. Let τ i : Vi → Qi, i = 1, 2, be vector bundles with an extrastructure given only by [ , ], ω satisfying conditions (a), (b) of Definition 6.1.1A.Let ρ : G × V → V be a vector bundle action. Assume that [ , ] or ω are invariantrelative to the action ρ, that is, condition (a) or (b) of Definition 6.1.6 holds. Thenthere are corresponding structures [ , ]G, [ω]G on the quotient vector bundle V/G.These quotient structures are naturally defined by:

(a) [[v1]G, [v2]G] = [[v1, v2]]G for all v1, v2 ∈ V such that τ(v1) = τ(v2), and

(b) [ω]G ([X]G, [Y ]G) = [ω(X,Y )]G for all X,Y ∈ TQ such that τQ(X) = τQ(Y ).

We will sometimes use the isomorphism αA of Lemma 2.4.2 as an identificationand, consequently, write [X]G ≡ Tπ(X)⊕ ξ, [Y ]G ≡ Tπ(Y )⊕ η. Thus, the secondidentity in the above lemma becomes

[ω]G ([X]G, [Y ]G)

≡ [ω]G(Tπ(X)⊕ ξ, Tπ(Y )⊕ η

)= [ω]G (Tπ(X), Tπ(Y )) + [ω]G (Tπ(X), η) + [ω]G

(ξ, Tπ(Y )

)+ [ω]G

(ξ, η).

Invariance Properties of Covariant Derivatives.

To pass covariant derivatives to quotient bundles, we will need some preparatorylemmas.

Lemma 6.1.8. Let τ : V → Q be a vector bundle and let D/Dt be a covariantderivative along curves associated to an affine connection ∇ on V (see the para-graphs following Definition 2.3.1 and Lemma 2.3.2). Let ρ : G×V → V be a vectorbundle action. If ρ commutes with D in the sense of Definition 6.1.6(c), then ∇ isinvariant in the following sense:

∇g∗Xg∗v = g∗∇Xv,

for all X ∈ X∞(Q), all v ∈ Γ(V ), and all g ∈ G. Conversely, if ∇ is invariant thenD/Dt is invariant.

The proof of this lemma is straightforward. See also the proof of Lemma 6.1.11(b)for a similar argument.


Horizontal and Vertical Components of Connections.

To define the notion of quotient covariant derivative or quotient connection, we willneed some extra structure, namely, a principal connection on Q. Our first job isto synthesize this connection with a given connection on a vector bundle to obtainhorizontal covariant derivatives and connections.

Definition 6.1.9. Let τ : V → Q be a vector bundle and let D/Dt be the covariantderivative along curves associated to a connection ∇ on V . Let ρ : G × V → V bea vector bundle action covering the action ρ0 : G × Q → Q. Let A be a principalconnection on the principal G-bundle Q→ Q/G (relative to the action ρ0). Let v(t)be any curve in V and let q(t) = τ (v(t)) for all t. Choose t0 and let q0 = q(t0). Letgq(t) and qh(t) be as in §2.2; that is, qh(t) is a horizontal curve, q(t) = gq(t)qh(t),and gq(t0) = e. Then we define

vh(t) = g−1q (t)v(t),

D(A,H)v(t)Dt

∣∣∣∣t=t0

=Dvh(t)Dt

∣∣∣∣t=t0

,

andD(A,V )v(t)

Dt

∣∣∣∣t=t0

=Dv(t)Dt

∣∣∣∣t=t0

− D(A,H)v(t)Dt

∣∣∣∣t=t0

.

We will callD(A,H)v(t)

Dt

the A-horizontal covariant derivative of v(t) and

D(A,V )v(t)Dt

the A-vertical covariant derivative of v(t). For X ∈ X∞(Q) and v ∈ Γ(V ) wealso define ∇(A,H)

X v and ∇(A,V )X v by

∇(A,H)X v(q0) =

D(A,H)

Dtv(t)

∣∣∣∣t=t0

and

∇(A,V )X v(q0) =

D(A,V )

Dtv(t)

∣∣∣∣t=t0

,

where the covariant derivatives on the right hand side are taken along any smoothcurve q(t) in Q satisfying q(t0) = q0, q(t0) = X(q0), and v(t) = v (q(t)) for allt. We will call ∇(A,H) the A-horizontal component and ∇(A,V ) the A-verticalcomponent of the connection ∇.


We see from this definition that

∇Xv(q0) = ∇(A,H)X v(q0) +∇(A,V )

X v(q0).

The following lemma gives, in particular, an alternative characterization of∇(A,H)X v

and ∇(A,V )X v for a G-invariant section v of V . It also shows that, when restricted

to invariant sections v ∈ ΓG(V ), the operator ∇(A,H) has the formal properties ofa connection.

Lemma 6.1.10. (a) Let HorA(X) ≡ XH and VerA(X) ≡ XV , for short. Thenwe have, for each q0 ∈ Q, each X ∈ X∞(Q), and each v ∈ ΓG(V ),

∇(A,H)X v(q0) = ∇XHv(q0)

and∇(A,V )X v(q0) = ∇XV v(q0).

In particular,∇Xv(q0) = ∇XHv(q0) +∇XV v(q0).

(b) Let v ∈ ΓG(V ) and let q(t) be any curve in Q such that q(t0) = X(q0). Defineqh(t) and gq(t) as in Definition 6.1.9 (see also §2.1). Then

∇(A,H)X v(q0) =

D

Dtg−1q (t)v(t)

∣∣∣∣t=t0

=D

Dtv(qh(t))

∣∣∣∣t=t0

and

∇(A,V )X v(q0) =

D

Dtgq(t)v(t0)

∣∣∣∣t=t0

.

In particular, ∇(A,V )X v(q0) depends only on ξ0 = gq(t0) and v(q0) = v(t0).

(c) Let v ∈ ΓG(V ), let q(t) be any curve in Q, and let v(t) = v (q(t)) for all t.Then

D(A,H)

Dtv(t)

∣∣∣∣t=t0

=D

Dtg−1q (t)v(t)

∣∣∣∣t=t0

=D

Dtv(qh(t))

∣∣∣∣t=t0

andD(A,V )

Dtv(t)

∣∣∣∣t=t0

=D

Dtgq(t)v(t0)

∣∣∣∣t=t0

.

Proof. (a) Let q(t) be any curve in Q such that q(t0) = X(q0) and let v(t) =v (q(t)) for all t. Then we can easily see that τ (vh(t)) = qh(t) for all t. Since v isG-invariant we have v(t) = gq(t)v (qh(t)). This argument and the definition of ∇,∇(A,H), and ∇(A,V ) imply the formulas in part (a) of the lemma.

(b) This part is basically a restatement of part (a). The first equality is a directconsequence of the definition of ∇(A,H). Now Let ξ0 = gq(t0). Then

XV (q0) = ξ0q0 =d

dt(gq(t)q0)

∣∣∣∣t=t0

.


Therefore the second equality is a consequence of the second equality of part (a)and the G-invariance of v.

(c) To prove this part we can proceed essentially as in the proof of parts (a) and(b). Alternatively, we can deduce it directly from part (b).

As a technical point, we note that in the proof of the previous lemma we arestrongly using the fact that the covariant derivative D/Dt is defined by a connection∇ which, in particular, has the formal property ∇X+Y = ∇X +∇Y .

The next two lemmas establish some basic properties of invariant covariantderivatives and invariant connections.

Lemma 6.1.11. (a) Assume the same hypothesis as in Definition 6.1.9 and, inaddition, that D/Dt is G-invariant in the sense of Definition 6.1.6(c). Then

D(A,H)

Dtand

D(A,V )

Dt

are also G-invariant in the following sense:

D(A,H)gv(t)Dt

= gD(A,H)v(t)

Dt

andD(A,V )gv(t)

Dt= g

D(A,V )v(t)Dt

,

for all curves v(t) in V and all g ∈ G.

(b) Let ∇(A,H) and ∇(A,V ) be the A-horizontal and A-vertical components of agiven connection ∇ on V and assume that ∇ is G-invariant. Then ∇(A,H)

and ∇(A,V ) are also G-invariant in the following sense:

∇(A,H)g∗X

g∗v = g∗∇(A,H)X v

and∇(A,V )g∗X

g∗v = g∗∇(A,V )X v,

for all X ∈ X∞(Q), all v ∈ Γ(V ), and all g ∈ G.

Proof. (a) Let a ∈ G and v(t) be given. Let τ (v(t)) = q(t). Then we immediatelyhave τ (av(t)) = aq(t). For any chosen t0 we have aq(t) = gaq(t)(aq)h(t) and q(t) =gq(t)qh(t) with gaq(t0) = gq(t0) = e. We know that (aq)h(t) = aqh(t) for all t. Usingthese equalities we can easily see that gaq(t) = agq(t)a−1 for all t. Then

D(A,H)

Dtav(t)

∣∣∣∣t=t0

=D

Dtg−1aq (t)av(t)

∣∣∣∣t=t0

=D

Dtag−1q (t)a−1av(t)

∣∣∣∣t=t0

=D

Dtag−1q (t)v(t)

∣∣∣∣t=t0

=D

Dtavh(t)

∣∣∣∣t=t0

= aD

Dtvh(t)

∣∣∣∣t=t0

= aD(A,H)

Dtv(t)

∣∣∣∣t=t0

.


To prove thatD(A,V )av(t)

Dt= a

D(A,V )v(t)Dt

for all curves v(t) on V and all a ∈ G we simply use Definition 6.1.9 and invarianceof D/Dt and of D(A,H)/Dt.

(b) To prove invariance of ∇(A,H) and ∇(A,V ) we use part (a) and the definitionsof ∇(A,H) and ∇(A,V ).

To prove part (b) of the preceding lemma for invariant v, we can use, alternatively,Lemma 6.1.10 and recall that for any g ∈ G and any X ∈ X∞(Q) we have (g∗X)H =g∗XH and also (g∗X)V = g∗XV .

Lemma 6.1.12. Assume the same hypothesis as in Lemma 6.1.11. Then for anycurve a(t) on G and any curve v(t) in V we have

D(A,H)

Dt[a(t)v(t)] = a(t)

D(A,H)

Dtv(t).

Proof. If q(t) = τ (v(t)), then a(t)q(t) = τ (a(t)v(t)). Choose t0. Then we have,for all t, gaq(t)(aq)h(t) = a(t)q(t) and also gq(t)qh(t) = q(t), where gaq(t0) =gq(t0) = e. We have (aq)h(t) = a(t0)qh(t) for all t, since both curves are horizontal,both pass through the same point a(t0)q(t0) at t = t0, and both project to the samecurve in Q/G. Therefore a(t)q(t) = gaq(t)a(t0)qh(t), for all t. On the other hand,a(t)q(t) = a(t)gq(t)qh(t) which then implies that gaq(t) = a(t)gq(t) (a(t0))−1. Usingthis and the G-invariance of D/Dt we obtain

D(A,H)

Dta(t)v(t)

∣∣∣∣t=t0

=D

Dta(t0) (gq(t))

−1a−1(t)a(t)v(t)

∣∣∣∣t=t0

= a(t0)D

Dt(gq(t))

−1v(t)

∣∣∣∣t=t0

= a(t0)D(A,H)

Dtv(t)

∣∣∣∣t=t0

.

Two More Properties of Covariant Derivatives.

The next two lemmas are included for the sake of completeness only, although theywill not be used in an essential way in the present work.

Lemma 6.1.13. Assume the same conditions as in Definition 6.1.9. Let v(t) bea curve in V , q(t) = τ (v(t)), and let f(t) be a real valued function. Then

D(A,H)

Dt[f(t)v(t)] = f(t)v(t) + f(t)

D(A,H)

Dtv(t)

6.2 Reduction of Lagrange–Poincare Bundles 75

for all t andD(A,V )

Dt[f(t)v(t)] = f(t)

D(A,V )

Dtv(t)

for all t.

Proof. It is not difficult to prove this lemma from the definitions. We will omitthe details.

The following lemma generalizes Lemma 6.1.12

Lemma 6.1.14. For each curve q(t) in Q and each choice of t0 define ξ(t0) =gq(t0). Thus, as t0 varies, ξ(t0) describes a curve in g which we will denote ξq(t).Assume the same conditions as in Definition 6.1.9. Now let v(t) be a curve in Vand let q(t) = τ (v(t)) for all t. Let a(t) be any curve in G. Then we have

(a)D(A,H)

Dt[a(t)v(t)]

∣∣∣∣t=t0

=D

Dt

[a(t0) (gq(t))

−1v(t)

]∣∣∣∣t=t0

(b)ξaq(t) = Ada(t) ξq(t) + a(t)a−1(t)

for all t. For each t,D(A,V )

Dt[a(t)v(t)]

only depends on a(t)v(t) and ξaq(t).

Proof. (a) The proof of this part is an immediate consequence of the last partof the proof of Lemma 6.1.12.

(b) From the proof of Lemma 6.1.12 it follows that gaq(t) = a(t)gq(t) (a(t0))−1.Using this, we can easily prove that ξaq(t) = Ada(t) ξq(t) + a(t)a−1(t).

6.2 Reduction of Lagrange–Poincare Bundles

Now we embark on the task of reducing Lagrange–Poincare bundles. The goal is tosee how the geometric structures on these bundles pass to the quotient bundle sothat the result is still a Lagrange–Poincare bundle. Most of this structure passesto the quotient in a fairly straightforward way. However, things are not as simplewith the covariant derivative, so we will need to pay special attention to this.

Now we are ready to pass covariant derivatives as well as horizontal and verticalcovariant derivatives and connections to the quotient.


The notion of quotient connection may be defined without any reference to quo-tient covariant derivatives. However, we shall treat these two notions together, whichwill be helpful in our later applications to variational principles.

There are some interesting links between the constructions here and those thatappear in the theory of geometric phases (see Marsden, Montgomery and Ratiu(1990), §13), but this will not be pursued in this paper.

Quotient Horizontal Covariant Derivatives and Connections.

Definition 6.2.1. Assume that the conditions of Definition 6.1.9 hold and, inaddition, assume that D/Dt is G-invariant; thus the associated connection ∇ onthe vector bundle V is also G-invariant.

(a) We define the quotient, or reduced, horizontal covariant derivative onthe vector bundle V/G by[

D(A,H)

Dt

]G

[v(t)]G =[D(A,H)

Dtv(t)

]G

for any curve v(t) on V . (This expression is well defined in view of Lemma6.1.11a.)

(b) We define the quotient, or reduced, horizontal connection by([∇(A,H)

]G

)[X]G

[v]G ≡[∇(A,H)

]G,[X]G

[v]G =[∇(A,H)X v

]G

for given [X]G ∈ Γ(TQ/G) and [v]G ∈ Γ(V/G) and corresponding X ∈ΓG(TQ) and v ∈ ΓG(V ). (This expression is well defined in view of Lemma6.1.11b.)

In this definition, recall that we identify elements of ΓG(TQ) with elements ofΓ(TQ/G) and also elements of ΓG(V ) with elements of Γ(V/G) using Lemma 5.2.2.

The connection associated to[D(A,H)/Dt

]G

is precisely[∇(A,H)

]G

in view ofLemmas 6.1.12 and 6.1.11(b).

The reduced horizontal connection can be naturally interpreted as a connection,which we shall denote ∇, on the vector bundle V/G in the usual sense. Usingthe general considerations in the paragraph Affine Connections in §2.3, this isexplained as follows. Let Y be any vector field on Q/G and let Y h be its horizontallift to Q. For any given [v]G ∈ Γ(V/G) define

∇Y [v]G :=[∇(A,H)

Y hv]G.

Then one can check that the properties defining a connection hold for ∇. Moreover,we can check that if X is any vector field such that HorA(X) = Y h, then

∇Y [v]G =[∇(A,H)X v

]G.


Quotient Vertical Covariant Derivatives and Connections.

It is important for us to define a quotient vertical covariant derivative as well as aquotient vertical connection. However, these will not be a covariant derivative anda connection in the usual sense.

Let us begin with a simple case, namely, the case of a constant curve in V/G.This is instructive because the answer turns out to be nonzero.

Let [v0]G ∈ V/G be given. Choose v0 ∈ V and q0 ∈ Q such that τ(v0) = q0. Letx0 ∈ Q/G be given by x0 = (τ/G) ([v0]G) = π(q0), where, as before, π : Q→ Q/Gis the projection. Let ξ0 ∈ g and consider the class [q0, ξ0]G ∈ gx0 .

Consider [v0]G as a constant curve; we define the notion of the vertical covariantderivative of this curve with respect to [q0, ξ0]G, as follows. Choose a curve g0(t) inG such that g0(t0) = e and g0(t0) = ξ0. Let v0(t) = g0(t)v0. Define the verticalcovariant derivative by[

D(A,V )

Dt

]G,[q0,ξ0]G

[v0]G

∣∣∣∣∣t=t0

=

[D(A,V )v0(t)

Dt

∣∣∣∣t=t0

]G

. (6.2.1)

Now we have to show that this expression is well defined. First of all, we need toexplain why the left hand side has the new subscript [q0, ξ0]G. This notation meansthat the class on the right hand side depends on the class [q0, ξ0]G. The reason forthis is as follows: during the quotient process, one represents the class [v0]G by anelement of V , say g(t)v0; the point is that the representative depends on a groupelement and this group element could be time dependent without changing the class[v0]G. The class of the resulting covariant derivative then depends on the class ofg, so this dependence on [q0, ξ0]G cannot be eliminated.

We now examine this in more detail, still using the constant curve [v0]G in thequotient. First of all, note that from Definition 6.1.9, and the fact that the quotientcurve x0 in Q/G is constant,

D(A,H)v0(t)Dt

∣∣∣∣t=t0

= 0

and therefore,D(A,V )v0(t)

Dt

∣∣∣∣t=t0

=Dv0(t)Dt

∣∣∣∣t=t0

.

It is clear that the right-hand side does not depend on the choice of the curve g0(t)satisfying g0(t0) = e, g0(t0) = ξ0, and now we shall see that it depends only on[v0]G ∈ V/G and [q0, ξ0]G ∈ gx0 and not on the particular choice of v0 and q0.Let us choose v1 ∈ [v0]G and also q1 ∈ [q0]G such that τ(v1) = q1. There is anh ∈ G such that q1 = hq0. Let ξ1 = Adh ξ0, and note that [q0, ξ0]G = [q1, ξ1]G.Using Lemma 5.2.1, we get v1 = hv0. If a curve g0(t) on G satisfies g0(t0) = e andg0(t0) = ξ0 then the curve g1(t) = hg0(t)h−1 satisfies g1(t0) = e and g1(t0) = ξ1.Define v1(t) = g1(t)v1. Then, v1(t) = hg0(t)h−1hv0 = hv0(t). Then using Lemma


6.1.11, we obtain[D(A,V )v1(t)

Dt

∣∣∣∣t=t0

]G

=

[hD(A,V )v0(t)

Dt

∣∣∣∣t=t0

]G

=

[D(A,V )v0(t)

Dt

∣∣∣∣t=t0

]G

.

Thus, we can consistently define the quotient vertical covariant derivative of theconstant curve [v0]G on V/G with respect to [q0, ξ0]G by equation (6.2.1).

Next we consider the case of general curves. Let [v]G(t) be a curve in V/G andx(t) = (τ/G) ([v]G(t)]G). Choose [q0, ξ0]G ∈ gx0 , where x0 = x(t0) = (τ/G) ([v]G(t0)).We want to define the covariant derivative of [v]G(t) with respect to [q0, ξ0]G att = t0. Choose v0 ∈ V and q0 ∈ Q such that τ(v0) = q0. Choose a curve g0(t) onG such that g0(t0) = e and g0(t0) = ξ0. Lemma 5.2.1 shows that there is a uniquecurve vhq0(t) in V such that

[vhq0(t)

]G

= [v]G(t) and τ(vhq0(t)

)= xhq0(t) for all t. Let

v0(t) = g0(t)vhq0(t) and q0(t) = g0(t)xhq0(t). Again by Lemma 5.2.1, one sees thatv0(t) is the only curve in V such that [v0(t)]G = [v]G(t) and τ (v0(t)) = q0(t) for allt. Then it is easy to see that

D(A,H)v0(t)Dt

∣∣∣∣t=t0

=D(A,H)vh0 (t)

Dt

∣∣∣∣t=t0

=Dvh0 (t)Dt

∣∣∣∣t=t0

does not depend on the choice of [q0, ξ0]G as before, although, of course, it dependson the choice of v0 and q0, satisfying τ(v0) = q0. Then, we have

D(A,V )v0(t)Dt

∣∣∣∣t=t0

=Dv0(t)Dt

∣∣∣∣t=t0

− Dvh0 (t)Dt

∣∣∣∣t=t0

.

It is clear that the right-hand side does not depend on the choice of the curve g0(t)satisfying g0(t0) = e and g0(t0) = ξ0.

Now we claim that [Dv0(t)/Dt]G depends only on the curve [v]G(t) in V/G and on[q0, ξ0]G ∈ gx0 but not on the particular choice of v0 and q0 satisfying the previousassumptions. Since we already know that [D(A,H)v0(t)/Dt]G depends only on thecurve [v]G(t) on V/G, we will then conclude, by definition of [D(A,V )v0(t)/Dt]G,that [D(A,V )v0(t)/Dt]G depends only on the curve [v]G(t) in V/G and on [q0, ξ0]G ∈gx0 but not on the particular choice of v0 and q0 satisfying the previous assumptions.

To prove the claim, choose v1 ∈ [v0]G and also q1 ∈ [q0]G such that τ(v1) = q1.Then there exists h ∈ G such that q1 = hq0 and ξ1 = Adh ξ0, and we also knowfrom Lemma 5.2.1 that v1 = hv0. If a curve g0(t) on G satisfies g0(t0) = e andg0(t0) = ξ0, let g1(t) = hg0(t)h−1, which satisfies, g1(t0) = e and g1(t0) = ξ1.Lemma 5.2.1 shows that there is a unique curve vhq1(t) in V such that

[vhq1(t)

]G

=[v]G(t) and τ

(vhq1(t)

)= xhq1(t) for all t. Define v1(t) = g1(t)vhq1(t) for all t and

also q1(t) = g1(t)xhq1(t). Then we see that v1(t) = hg0(t)h−1hv0 = hv0(t). Lemma5.2.1 implies that v1(t) is the only curve in V such that [v1(t)]G = [v]G(t) andτ (v1(t)) = q1(t) for all t. Then using Lemma 6.1.11, we obtain[

Dv1(t)Dt

∣∣∣∣t=t0

]G

=

[hDv0(t)Dt

∣∣∣∣t=t0

]G

=

[Dv0(t)Dt

∣∣∣∣t=t0

]G

.


This and the definition of the vertical covariant derivative shows that[D(A,V )v1(t)

Dt

∣∣∣∣t=t0

]G

=

[hD(A,V )v0(t)

Dt

∣∣∣∣t=t0

]G

=

[D(A,V )v0(t)

Dt

∣∣∣∣t=t0

]G

.

This argument also shows that we can define the quotient covariant derivativeof the curve [v]G(t) in V/G with respect to [q0, ξ0]G by[

D(A)

Dt

]G,[q0,ξ0]G

[v]G(t)

∣∣∣∣∣t=t0

=

[Dv0(t)Dt

∣∣∣∣t=t0

]G

.

This justifies the following definition of the reduced covariant derivative and alsothe reduced vertical covariant derivative.

Definition 6.2.2. Assume the same conditions as in Definition 6.1.9 and assumethat the connection ∇, and therefore also the covariant derivative D/Dt, is G-invariant. Let [v]G(t) be a given curve in V/G. Let (τ/G)[v]G(t0) = [q0]G and choose[q0, ξ0]G ∈ gx0 . Let vhq0(t) be the unique curve in V such that

[vhq0(t)

]G

= [v]G(t)and τ

(vhq0(t)

)= xhq0(t) for all t. Choose a curve g0(t) on G such that g0(t0) = e

and g0(t0) = ξ0. Let v0(t) = g0(t)vhq0(t).

(a) The quotient, or reduced, covariant derivative of the curve [v]G(t) withrespect to [q0, ξ0]G at t = t0 is defined by[

D(A)

Dt

]G,[q0,ξ0]G

[v]G(t)

∣∣∣∣∣t=t0

=

[Dv0(t)Dt

∣∣∣∣t=t0

]G

.

(b) The quotient, or reduced, vertical covariant derivative of the curve[v]G(t) with respect to [q0, ξ0]G at t = t0 is defined by[

D(A,V )

Dt

]G,[q0,ξ0]G

[v]G(t)

∣∣∣∣∣t=t0

=

[D(A,V )v0(t)

Dt

∣∣∣∣t=t0

]G

.

The following formula is a direct consequence of the previous definition andprevious equalities[

D(A)

Dt

]G,[q0,ξ0]G

[v]G(t)

∣∣∣∣∣t=t0

=

[D(A,V )

Dt

]G,[q0,ξ0]G

[v]G(t)

∣∣∣∣∣t=t0

+[D(A,H)

Dt

]G

[v]G(t)∣∣∣∣t=t0

.

Next we consider an important particular case in which the reduced verticalcovariant derivative can be calculated with a formula entirely similar to the onederived before for the case of a constant curve.


Lemma 6.2.3. Assume the conditions of definition 6.2.2 and, in addition, assumethat there is a G-invariant section u ∈ ΓG (V ) such that v0(t) = u

(g0(t)xhq0(t)(t)

).

Then [D(A,V )

Dt

]G,[q0,ξ0]G

[v]G(t)

∣∣∣∣∣t=t0

=

[Dg0(t)v0

Dt

∣∣∣∣t=t0

]G

.

Proof. The proof is a direct consequence of Lemma 6.1.10(c).

Quotient Vertical Connections.

Now we will describe the notion of quotient, or reduced, vertical connection andalso the quotient, or reduced, connection.

Definition 6.2.4. Let [q0, ξ0]G ∈ g with πG[q0, ξ0]G = [q0]G and let [v]G ∈Γ(V/G), where v ∈ ΓG(V ) according to Lemma 5.2.2. Let Y0 ∈ IVG (Q) such thatβA(Y0) = [q0, ξ0]G, according to Lemma 5.1.2. Then

(a) The quotient, or reduced, vertical connection is defined by[∇(A,V )

]G,[q0,ξ0]G

[v]G =[∇(A,V )Y0

v]G.

(b) Let X0 ∈ T[q0]G(Q/G), define X0 = Xh0 , so X0 ∈ IHG (Q), and Z0 = X0 + Y0.

The quotient, or reduced, connection is defined by the condition[∇(A)

]G,X0⊕[q0,ξ0]G

[v]G = [∇Z0v]G ,

or by the equivalent condition[∇(A)

]G,X0⊕[q0,ξ0]G

[v]G =[∇(A,H)

]G,X0

[v]G +[∇(A,V )

]G,[q0,ξ0]G

[v]G

One sees that the expressions in Definition 6.2.4 are well defined using Lemma6.1.11, which establishes invariance of ∇(A,V ) and ∇(A,H), and also Lemma 5.2.2.

The following lemma establishes the link between the notions of quotient verti-cal covariant derivative (resp. quotient covariant derivative) and quotient verticalconnection (resp. quotient connection).

Lemma 6.2.5. Let [q0, ξ0]G ∈ g and let [v]G ∈ Γ(V/G), where v ∈ ΓG(V )according to Lemma 5.2.2. Let [q]G(t) = x(t) be any curve in Q/G such thatπG ([q0, ξ0]G) = [q]G(t0) and let, with a convenient abuse of notation, [v]G(t) =[v]G ([q]G(t)). Assume that D/Dt is G-invariant. Then we have

(a) [D(A,V )

Dt

]G,[q0,ξ0]G

[v]G

∣∣∣∣∣t=t0

=[∇(A,V )

]G,[q0,ξ0]G

[v]G ([q]G(t0))


(b) [D(A)

Dt

]G,[q0,ξ0]G

[v]G

∣∣∣∣∣t=t0

=[∇(A)

]G,x(t0)⊕[q0,ξ0]G

[v]G ([q]G(t0)) .

Proof. The proof is a direct consequence of Lemma 6.2.3 and the fundamentallink between covariant derivatives and connections in a vector bundle explainedbefore (see the paragraph Affine Connections in §2.3).

Group Actions on Lagrange–Poincare Bundles.

Now we shall define the notion of an action of a group G on an object of LP.

Definition 6.2.6. An action in the category LP of a group G on an objectTQ⊕ V of LP is a vector bundle action ρ : G× TQ⊕ V → TQ⊕ V such that, foreach g ∈ G, ρg : TQ⊕ V → TQ⊕ V is an isomorphism of LP. Sometimes we willcall an action in the category LP simply an action, if the fact that it is an actionin the category LP is clear from the context.

Corollary 6.2.7. If ρ : G× TQ⊕ V → TQ⊕ V is an action in the category LPthen we have

(i) ρ∗ : G × Γ(TQ ⊕ V ) → Γ(TQ ⊕ V ) is a representation of G such that, foreach g ∈ G, ρg∗ : Γ(TQ⊕ V )→ Γ(TQ⊕ V ) is a Lie algebra isomorphism;

(ii) The restriction ρ|G× TQ : G× TQ→ TQ is the tangent lift of the action onthe zero section, ρ0 : G×Q→ Q.

Proof. (i) is a consequence of Lemma 6.1.5 and (ii) is a consequence of the defi-nition of morphism in the category LP.

Definition 6.2.8. Let τ : V → Q be a vector bundle and let ρ : G × V → V bea vector bundle action. As usual, the action on the zero section ρ0 : G×Q→ Q isassumed to give Q the structure of a principal bundle. Let ιG(V ) : IG(V )→ Q/G bethe vector bundle whose fiber at the point [q]G ∈ Q/G is the space of all G-invariantsections of the restriction V |Gq, where, as usual, Gq = [q]G is the orbit of q. Inaddition, let γG(V ) be the vector bundle isomorphism γG(V ) : IG(V )→ V/G givenby

γG(V )(v[q]G

)=[v[q]G(q)

]G,

where v[q]G is a section of V |Gq.

It is easy to see that γG(V ) is a well defined vector bundle isomorphism. Observethat Definition 6.2.8 is consistent with Definition 5.1.1; in fact, we can easily seethat IHG (TQ) = IG (Hor(TQ)) and IVG (TQ) = IG (Ver(TQ)). Obviously we have avector bundle isomorphism

γ : IHG (TQ)⊕ IVG (TQ)⊕ IG(V )→ Hor(TQ)/G⊕Ver(TQ)/G⊕ (V/G)


defined by

γ ≡ γG (Hor(TQ))⊕ γG (Ver(TQ))⊕ γG (V )≡ γG (Hor(TQ)⊕Ver(TQ)⊕ V ) .

Isomorphisms Between Quotient Bundles.

Using the definition of βA from Lemma 5.1.2, taking into account the natural iden-tifications TQ ≡ Hor(TQ)⊕Ver(TQ) and also TQ/G ≡ Hor(TQ)/G⊕Ver(TQ)/G,and using the isomorphism γG(V ) just defined, we obtain a vector bundle isomor-phism

βA ⊕ γG(V ) : IHG (TQ)⊕ IVG (TQ)⊕ IG(V )→ T (Q/G)⊕ g⊕ (V/G).

Also from Lemma 5.2.2 we see that there are linear isomorphisms

πG(V )∗ : Γ(V/G)→ ΓG(V ),

πG(Hor(TQ))∗ : Γ (Hor(TQ)/G)→ ΓG (Hor(TQ)) ,

andπG(Ver(TQ))∗ : Γ (Ver(TQ)/G)→ ΓG (Ver(TQ)) .

Definition 6.2.9. Define the linear isomorphism

π∗ : ΓG (Hor(TQ))⊕ ΓG (Ver(TQ))⊕ ΓG(V )→ Γ (Hor(TQ)/G)⊕ Γ (Ver(TQ)/G)⊕ Γ(V/G)

byπ∗ = (πG(Hor(TQ))∗ ⊕ πG(Ver(TQ))∗ ⊕ πG(V )∗)−1

.

Similarly, the linear isomorphism

ϕ ≡ ϕA(Q,G, V ) ≡ ϕA : ΓG (Hor(TQ))⊕ ΓG (Ver(TQ))⊕ ΓG(V )→ Γ (T (Q/G))⊕ Γ (g)⊕ Γ(V/G)

is defined byϕ = (βA ⊕ γG(V ))∗ γ

∗ π∗.

If we consider γ∗ π∗ as being an identification, which is natural in the presentcontext, we can write ϕ ≡ (βA ⊕ γG(V ))∗. Likewise, if we consider γ∗ and π∗ asbeing identifications we can write the map

ϕ : Γ(TQ/G)⊕ Γ(V/G)→ Γ (T (Q/G)⊕ g)⊕ Γ(V/G)

as ϕ ≡ α∗A ⊕ id(V/G)∗.


Structure of Reduced Lagrange–Poincare Bundles.

Now we state and prove our main result in this section.

Theorem 6.2.10. Let τQ ⊕ τ : TQ ⊕ V → Q be an object of LP and let ∇, ω,[ , ] be the structure on V . Let ρ : G × (TQ ⊕ V ) → TQ ⊕ V be an action in thecategory LP. Choose a connection A on the principal bundle Q with structure groupG. Consider the bundle

T (Q/G)⊕ g⊕ (V/G).

Define the structure ∇g, ωg, [ , ]g on g⊕ (V/G) by

∇gX

(ξ ⊕ [v]G

)= ∇AX ξ ⊕ [∇(A,H)]G,X [v]G − [ω]G(X, ξ)

ωg(X1, X2) = BA(X1, X2)⊕ [ω]G(X1, X2)[ξ1 ⊕ [v1]G, ξ2 ⊕ [v2]G

]g = [ξ1, ξ2]⊕ [∇(A,V )]G,ξ1 [v2]G − [∇(A,V )]G,ξ2 [v1]G− [ω]G(ξ1, ξ2) + [[v1]G, [v2]G] .

Then the bundle T (Q/G)⊕ g⊕ (V/G), with the structure on the bundle g⊕ (V/G)given by ∇g, ωg, [ , ]g, is an object of the category LP.

Remark.

Theorems 6.2.10 and 5.2.4 seem to shed light on the structure of Lie group ex-tensions, namely that they necessarily have the structure, roughly speaking, of asemidirect product with a curvature cocycle. It would be of interest to explore thistechnical point further.

Proof of the Theorem.. It is easy to see that ∇g is a connection on g⊕ (V/G)(see the paragraph Affine Connections of §2.3 for the definition). It is also easy tocheck that ωg is a g⊕(V/G) -valued 2-form. We must now show that the expressionof [ , ]g given in the statement of the theorem defines a Lie bracket on the fibers ofg ⊕ (V/G) that endows this bundle with the structure of a Lie algebra bundle. Itis clear that each one of the terms [ξ1, ξ2], −[ω]G(ξ1, ξ2) and [[v1]G, [v2]G]] definesa bilinear and skew symmetric operation on the fibers of g ⊕ (V/G). Now usingLemma 6.1.10 and Definition 6.2.4 we can see that the term

[∇(A,V )]G,ξ1 [v2]G − [∇(A,V )]G,ξ2 [v1]G

also defines a bilinear and skew symmetric operation on the fibers of g⊕ (V/G). Sofar we have proven that the expression of [ , ]g defines a bilinear and skew symmetricoperation on the fibers of g⊕(V/G). Moreover, it is easy to see that for any functionf ∈ C∞(Q/G) we have[

ξ1 ⊕ [v1]G, f(ξ2 ⊕ [v2]G

)]g = f[ξ1 ⊕ [v1]G, ξ2 ⊕ [v2]G

]g.

It remains to prove that [ , ]g satisfies the Jacobi identity. For this we must firststudy the Lie algebra structure on the space of sections

Γ (T (Q/G))⊕ Γ (g)⊕ Γ(V/G) ≡ Γ ((T (Q/G))⊕ g⊕ (V/G)) .


According to Definition 6.1.1 the operation on sections Zi ⊕ vi ∈ Γ(TQ ⊕ V ),i = 1, 2, given by

[Z1 ⊕ v1, Z2 ⊕ v2] = [Z1, Z2]⊕∇Z1v2 −∇Z2v1 − ω(Z1, Z2) + [v1, v2]

is a Lie bracket.The G-invariance of ∇, ω, and [ , ] implies that

ΓG(TQ⊕ V ) ≡ ΓG (Hor(TQ))⊕ ΓG (Ver(TQ))⊕ ΓG(V )

is a Lie subalgebra. Using this and the linear isomorphism ϕ introduced in Definition6.2.9, we can define a Lie algebra on the space of sections

Γ (T (Q/G))⊕ Γ (g)⊕ Γ(V/G)

by [X1 ⊕ ξ1 ⊕ [v1]G, X2 ⊕ ξ2 ⊕ [v2]G

]= ϕ

[ϕ−1

(X1 ⊕ ξ1 ⊕ [v1]G

),

ϕ−1(X2 ⊕ ξ2 ⊕ [v2]G

)].

We now show that this Lie bracket coincides with the one constructed on

Γ (T (Q/G))⊕ Γ (g)⊕ Γ(V/G)

using the structures ∇g, ωg and [ , ]g, according to Definition 6.1.1. We know thatϕ−1Xi = Xh

i , for i = 1, 2. Let ϕ−1ξi = Yi and ϕ−1[vi]G = vi, for i = 1, 2 (recallthat there is a unique vi ∈ ΓG(V ) in each class [vi]G). By Definition 6.1.1 we have

[Xh1 ⊕ Y1 ⊕ v1, X

h2 ⊕ Y2 ⊕ v2]

= [Xh1 + Y1, X

h2 + Y2]⊕∇Xh1 +Y1

v2 −∇Xh2 +Y2v1

− ω(Xh1 + Y1, X

h2 + Y2) + [v1, v2].

Since ϕ |ΓG(TQ) = αA∗ |ΓG(TQ) we have, using Theorem 5.2.4, that

ϕ[Xh1 ⊕ Y1, X

h2 ⊕ Y2] = [X1, X2]⊕ ∇AX1

ξ2 − ∇AX2ξ1 − BA(X1, X2) + [ξ1, ξ2].

Using Definition 6.2.4 we have, for

ϕ(∇Xhi +Yivj) =[∇(A,H)

]G,Xi

[vj ]G +[∇(A,V )

]G,ξi

[vj ]G.

Using Lemma 6.1.7 we can easily see that

ϕω(Xh1 + Y1, X

h2 + Y2)

= [ω]G(X1, X2) + [ω]G(X1, ξ2) + [ω]G(ξ1, X2) + [ω]G(ξ1, ξ2)

and also thatϕ[v1, v2] = [[v1]G, [v2]G] .


It is not difficult to conclude that the operation on

Γ ((T (Q/G))⊕ g⊕ (V/G))

using the structures ∇g, ωg, and [ , ]g, according to Definition 6.1.1 is a Lie bracket.Moreover, we have a description of this bracket using the isomorphism ϕ. It followsthat the restriction of this Lie bracket to Γ (g⊕ (V/G)) is also a Lie bracket. Thisrestriction is given by [ , ]g. Therefore, the Jacobi identity for the bilinear and skewsymmetric operation on fibers of Γ (g⊕ (V/G)) defined by [ , ]g is satisfied. Thisfinishes the proof of the theorem.

Now we introduce some more notation.

Definition 6.2.11. Assume the hypothesis of Theorem 6.2.10. The bundle

T (Q/G)⊕ g⊕ (V/G)

with the reduced structure ∇g, ωg, and [ , ]g on g⊕ (V/G) is called the reducedbundle with respect to the group G and the connection A. The reduced Liealgebra structure on

Γ(T (Q/G)⊕ g⊕ (V/G))

is the one described in Theorem 6.2.10 using the reduced structure ∇g, ωg, [ , ]g ong⊕ (V/G) according to Definition 6.1.1.

Let W = TQ⊕ V , for short. We will denote by αWA the natural map

αWA : (TQ⊕ V )/G→ T (Q/G)⊕ g⊕ (V/G),

that is, αWA := αA ⊕ idV/G, where αA is the map of Lemma 2.4.2. To make thenotation consistent in the case V = 0, that is, W = TQ, we will write αA ≡ αTQA .

Lemma 6.2.12. Assume the hypothesis of Theorem 6.2.10. Then we have:

(a) The push forward map

αWA∗ : Γ ((TQ⊕ V )/G)→ Γ (T (Q/G)⊕ g⊕ (V/G))

is a Lie algebra isomorphism. Here the Lie algebra on Γ ((TQ⊕ V )/G) is thenatural quotient Lie algebra structure, as defined in the comments followingLemma 5.2.2.

(b) Let A′ be another connection on the principal bundle Q with structure groupG. Then αWA′ (αWA )−1 is an isomorphism in the category LP if and only ifA = A′.

Proof. (a) The proof of this part is an easy consequence of the proof of Theorem6.2.10.

(b) LetX ≡ X ⊕ 0⊕ 0 ∈ Γ (T (Q/G)⊕ g⊕ (V/G)) .


The proof of Theorem 6.2.10 shows that (αWA )−1(X) is the horizontal lift of X withrespect to the connection A and hence (αWA′ (αWA )−1)(X) can be written as

(αWA′ (αWA )−1)(X) = X ⊕ ξ,

where ξ ∈ g. Also, by the same argument as in the proof of Theorem 6.2.10, we cansee that ξ = 0 if and only if (αWA )−1(X) is horizontal with respect to the connectionA′. On the other hand, the proof of Theorem 6.2.10, also shows that for any

ξ ⊕ [v]G ≡ 0⊕ ξ ⊕ [v]G ∈ Γ (T (Q/G)⊕ g⊕ (V/G))

we have(αWA′ (αWA )−1)

(ξ ⊕ [v]G

)∈ Γ (g⊕ (V/G)) .

Using what we have proven so far, we can deduce that the conditions(αWA′ (αWA )−1

)T (Q/G) = T (Q/G)

and (αWA′ (αWA )−1

)(g⊕ (V/G)) = g⊕ (V/G)

are satisfied if and only if A = A′.

6.3 Reduction by Stages of objects of LP

We shall begin by recollecting, without proof, some basic facts. In this section Gwill be a Lie group and N ⊂ G a normal Lie subgroup.

Lemma 6.3.1. Assume that a group G acts on the left on a manifold Q (this timewe do not need to assume that, with this action, Q becomes a principal bundle) andlet N ⊂ G be a normal subgroup. Then the rule [g]N [q]N = [gq]N defines an actionof G/N on Q/N . The map

iQG/N : Q/G→ (Q/N)/(G/N)

given byiQG/N ([q]G) = [[q]N ](G/N)

is a bijection.If the action of G on Q is free then the action of N on Q is free and also the

action of G/N on Q/N is free. Conversely, if the action of N on Q is free and alsothe action of G/N on Q/N is free then the action of G on Q is free.

Now assume, in addition, that π : Q→ Q/G, is a principal bundle with structuregroup G. Then the map iQG/N is a well defined diffeomorphism. Moreover, Q/Nis a principal bundle with structure group G/N and Q is a principal bundle withstructure group N . Conversely, if Q/N is a principal bundle with structure groupG/N and Q is a principal bundle with structure group N then Q is a principalbundle with structure group G.

6.3 Reduction by Stages of objects of LP 87

We are interested in cases of the previous lemma in which the manifold Q carriessome extra structures which remain invariant under the action of the group G andwe also want to show how to obtain the corresponding quotient structures by stages.The next lemma considers the case of an invariant vector bundle structure. We willuse some notation and results described in section §5.2, in particular Lemma 5.2.1

Lemma 6.3.2. Let τ : V → Q be a vector bundle and let ρ : G × V → V be avector bundle action of the group G on V covering the action ρ0 : G × Q → Q onthe zero section. We assume that the action ρ0 endows Q with the structure of aprincipal bundle over Q/G. Let N ⊂ G be a normal subgroup. Then

τ/N : V/N → Q/N and τ/G : V/G→ Q/G

are vector bundles and G/N acts with a vector bundle action on V/N . The quotient

(τ/N)/(G/N) : (V/N)/(G/N)→ (Q/N)/(G/N)

is a vector bundle. The isomorphism iVG/N is a vector bundle isomorphism.

Now we state the following lemma whose proof is straightforward.

Lemma 6.3.3. Assume the hypothesis of Lemma 6.1.7 and, in addition, assumethat N is a normal subgroup of G. Thus, in particular, any one of the structures(a) or (b) defined in Lemma 6.1.7 which is invariant under the action of G isalso invariant under the restricted action of N , so it gives rise to a correspondingquotient structure on the quotient vector bundle V/N . Then the quotient structuresso defined are invariant under the action of G/N on V/N described in Lemma 6.3.2and they define corresponding quotient structures on the bundle (V/N)/(G/N).Moreover, the isomorphism iVG/N described in Lemma 6.3.2 commutes with thosestructures, that is, it is also an isomorphism with respect to these structures.

Let us now consider an important example of this situation. Assume that thehypothesis of Lemma 6.3.1 hold. Let n be as in Definition 2.3.3. Then G/N acts onn as follows:

[g]N [q, ξ]N = [gq,Adg ξ]N .

We will prove that this action is well defined. Since N ⊂ G is a normal subgroup,for any g ∈ G and any n ∈ N there exist n′g, n

′′g ∈ N such that gn = n′gg and

ng = gn′′g . Besides, for any ξ ∈ n we have Adg ξ ∈ n. Now let ni ∈ N be given fori = 1, 2. Then we have

[n1g]N [n2q,Adn2 ξ]N = [n1gn2q,Adn1g Adn2 ξ]N= [n1n

′2ggq,Adn1n′2gg

ξ]N

= [n1n′2ggq,Adn1n′2g

Adg ξ]N

= [gq,Adg ξ]N ,

which shows that the action is well defined.Now we will study invariance properties of the structures on n. We will omit the

proof of the next lemma, which is straightforward.


Lemma 6.3.4. Assume the same hypothesis as in Lemma 6.3.3 and, in addition,choose a connection AN on the principal bundle Q with structure group N . Thenthe action of G/N on n defined above commutes with the Lie algebra structure onfibers of n defined in Lemma 2.3.5 and also with the n-valued curvature 2-formBAN defined by equation (3.1.1). More precisely, for any [q, ξi]N ∈ n, i = 1, 2, anyXi ∈ T[q]N (Q/N), i = 1, 2, and any g ∈ G we have

[[g]N [q, ξ1]N , [g]N [q, ξ2]N ] = [g]N [[q, ξ1]N , [q, ξ2]N ]

andBAN ([g]NX1, [g]NX2) = [g]N BAN (X1, X2).

It is not generally true that for any connection AN the covariant derivativeDAN /Dt or, equivalently, the connection ∇AN on the bundle n, is invariant underthe action of G/N . However there is always a connection AN having this property,that is, having the property that the action of G/N commutes with DAN /Dt or,equivalently, with the connection ∇AN , as we shall see next. Let 〈 , 〉 be any G-invariant metric on Q. It is not difficult to show that any given principal bundle,say π : Q→ Q/G with structure group G, carries an invariant Riemannian metric.For instance, it is well known that using partitions of unit one can construct both,a Riemannian metric g on the base Q/G and also a principal connection A on theprincipal bundle Q. With respect to the connection A, we have the decompositionTQ = HorTQ⊕VerTQ. We are going to define an invariant metric gh on the vectorbundle HorTQ and an invariant metric gv on the vector bundle VerTQ. Then, bydeclaring that HorTQ and VerTQ are orthogonal, g = gh⊕ gv will be an invariantpositive definite metric on the vector bundle TQ, that is, an invariant Riemannianmetric on Q. We can define gh as being the horizontal lift of the Riemannian metricg on the base. To define gv, let us recall that the vector bundle VerTQ is isomorphicto the vertical invariant bundle IVG (TQ), defined in 5.1.1. It is also well known thatany vector bundle carries a positive definite metric, so let gv be a positive definitemetric on the vector bundle IVG (TQ). Define the metric gv on VerTQ as follows: Foreach Xq, Yq ∈ VerTQ, let X,Y ∈ IVG (TQ) be such that X(q) = Xq and Y (q) = Yq,then set, by definition, gv(Xq, Yq) = gv(X,Y ).

Then for each q ∈ Q we have a direct sum orthogonal decomposition

TqQ = VerN (TqQ) +HN (q),

where HN (q) is the orthogonal complement of VerN (TqQ). Given vq ∈ TqQ we canthen write vq = ξq + vqHN where vqHN ∈ HN (q) and ξ ∈ n. We can easily see thatthe restriction

TqπN |HN (q) : HN (q)→ T[q]N (Q/N)

is a linear isomorphism and that the collection of all HN (q) is G-invariant, therefore,also N -invariant. Thus the collection of all HN (q) defines a connection on theprincipal bundleQ with structure groupN . LetAN be the corresponding connection1-form. By definition, we have AN (vq) = ξ if vq = ξq+vqHN . The horizontal spacesof this connection are HorANq (TQ) ≡ HN (q).


Lemma 6.3.5. Let AN be the connection associated to a G-invariant metric 〈 , 〉as explained above. Then for any g ∈ G and any vq ∈ TqQ we have AN (gvq) =Adg AN (vq).

Proof. For any g ∈ G and any vq ∈ TqQ we have

AN (gvq) = AN (gξq + gvqHN ) = AN (gξq) +AN (gvqHN ).

Since the collection of horizontal spaces is G-invariant, it follows that gvqHN ∈HN (gq) and therefore AN (gvqHN ) = 0. On the other hand we have AN (gξq) =AN (Adg ξgq). Since N is normal, we have Adg ξ ∈ n and we get

AN (gvq) = Adg ξ = Adg AN (vq) for all q ∈ Q.

Lemma 6.3.6. (a) Let AN be a connection on the principal bundle Q withstructure group N having the property that for any g ∈ G and any vq ∈ TqQwe have AN (gvq) = Adg AN (vq). Let ∇AN be the covariant derivative on n

associated to AN according to Definition 2.3.1. Then ∇AN is G/N -invariant.

(b) Assume the same conditions as in (a). Then the action of G/N on T (Q/N)⊕ndefined by the natural actions of G/N on T (Q/N) and on n commutes withthe isomorphism αAN : TQ/N → T (Q/N)⊕ n (see Lemma 2.4.2).

Proof. (a) Let DAN /Dt be the covariant derivative along curves associated tothe connection ∇AN . Using the previous lemmas, the proof of this part is straight-forward. Indeed, according to Lemma 2.3.4 we have

DAN [q(s), ξ(s)]GDs

=[q(s),− [AN (q(s), q(s)) , ξ(s)] + ξ(s)

]N.

Given any [g]N ∈ G/N we have

DAN [gq(s),Adg ξ(s)]GDs

=[gq(s),− [AN (gq(s), gq(s)) ,Adg ξ(s)] + Adg ξ(s)

]N

=[gq(s),−Adg [AN (q(s), q(s)) , ξ(s)] + Adg ξ(s)

]N

= [g]N[q(s),− [AN (q(s), q(s)) , ξ(s)] + ξ(s)

]N,

which proves the G/N -invariance of the covariant derivative DAN /Dt.(b) This follows easily from the previous lemmas.

Theorem 6.3.7. Let Q be a principal bundle with structure group G and let Nbe a normal subgroup of G. Let AN be a connection on the principal bundle Q withstructure group N having the property that for any g ∈ G and any vq ∈ TqQ wehave AN (gvq) = Adg AN (vq). Consider the action of G/N on T (Q/N)⊕n such that


its restriction to n coincides with the natural action and its restriction to T (Q/N)coincides with the tangent lift of the action of G/N on Q/N defined in Lemma6.3.1. Then this action is an action in the category LP. The reduced bundle ofT (Q/N)⊕ n by the group G/N with respect to any connection AG/N on the bundleQ/N with structure group G/N is an object of LP.

Proof. The proof is a direct consequence of Definition 6.2.4, Lemmas 6.3.4 and6.3.6, and Theorem 6.2.10.

More Preparatory Lemmas.

Before we can state and prove one of our main results on Lagrangian reduction bystages we need a few more lemmas.

Lemma 6.3.8. Assume the conditions of Theorem 6.3.7 and, in addition, choosea connection AG on the principal bundle Q with structure group G. Let K = G/N .Consider the following vector bundle isomorphisms:

(a) the quotient isomorphism (see Lemma 6.3.6 and Definition 6.2.11)

[αTQAN ]G/N : (TQ/N)/(G/N)→ (T (Q/N)⊕ n) /(G/N),

(b) the isomorphism (see Lemma 6.3.1)

iTQG/N : TQ/G→ (TQ/N)/(G/N),

(c) the isomorphism (see Definition 6.2.11)

αTQAG : TQ/G→ T (Q/G)⊕ g,

(d) the isomorphism (see Definition 6.2.11)

αT (Q/N)⊕nAG/N

: (T (Q/N)⊕ n) /(G/N)

→ T ((Q/N)/(G/N))⊕ k⊕ (n/(G/N)) ,

where k is the Lie algebra of K (and hence one has a Lie algebra isomorphismk ≡ g/n).

Then the linear isomorphisms, given by the push forward of maps corresponding tothe previous vector bundle isomorphisms

(a’)[αTQAN ]G/N∗ : Γ ((TQ/N)/(G/N))→ Γ ((T (Q/N)⊕ n) /(G/N))

(b’)iTQG/N∗ : Γ (TQ/G)→ Γ ((TQ/N)/(G/N))


(c’)αTQAG∗ : Γ (TQ/G)→ (T (Q/G)⊕ g)

(d’)

αT (Q/N)⊕nAG/N∗ : Γ ((T (Q/N)⊕ n) /(G/N))

→ Γ(T ((Q/N)/(G/N))⊕ k⊕ (n/(G/N))

)are Lie algebra isomorphisms.

Consider the composition

(e)εTQ(AN ,AG/N ,AG) = α

T (Q/N)⊕nAG/N

[αTQAN ]G/N iTQG/N (αTQAG )−1,

which is a vector bundle isomorphism from

T (Q/G)⊕ g onto T ((Q/N)/(G/N))⊕ k⊕ (n/(G/N)) .

Then εTQ(AN ,AG/N ,AG) induces a Lie algebra isomorphism

(e’)

εTQ(AN ,AG/N ,AG)∗ : Γ ((T (Q/G)⊕ g))

→ Γ(T ((Q/N)/(G/N))⊕ k⊕ (n/(G/N))

).

Proof. The conclusion follows using Lemmas 5.2.1, 6.3.6, 6.2.12, the paragraphon quotient Lie algebras in section §5.2, and standard results on quotient manifolds.

It is not generally true that εTQ(AN ,AG/N ,AG) is an isomorphism in the category LP,for arbitrary choices of the connections AN , AG/N , and AG. Next we will show thatit is always possible to choose AN , AG/N , and AG in such a way that εTQ(AN ,AG/N ,AG)

is an isomorphism in the category LP.

Lemma 6.3.9. Assume the hypothesis of Lemma 6.3.5 and define the connectionAN as in that lemma. Define the connection AG on the principal bundle Q withstructure group G by the condition that, for all q ∈ Q, HorAGq (TQ) is the orthogonalcomplement of VerGq (TQ). For each [q]N ∈ Q/N let

H[q]N (T (Q/N)) = TπN(HorAGq (TQ)

),

where πN : Q → Q/N is the natural principal bundle projection. Then the collec-tion of all H[q]N (T (Q/N)) as [q]N varies in Q/N defines a connection AG/N on


the principal bundle Q/N with structure group G/N , by choosing these to be thehorizontal spaces, that is,

HorAG/N[q]N((T (Q/N)) ≡ H[q]N (T (Q/N)) .

The following property is satisfied: for all vq ∈ TQ and all q ∈ Q we haveAG(vq) = 0 if and only if AN (vq) = 0 and AG/N (TπN (vq)) = 0.

We shall omit the proof of this lemma which can be carried out in a standardway using the preceding results.

The next theorem, together with its generalization for arbitrary objects of LPgiven in Theorem 6.3.14 constitutes one of our main results on Lagrangian reductionby stages.

Theorem 6.3.10. Assume the hypothesis of Theorem 6.3.7 and, in addition,choose the connections AN , AG/N , and AG in such a way that the following prop-erty is satisfied: for all vq ∈ TQ and all q ∈ Q we have AG(vq) = 0 if and only ifAN (vq) = 0 and AG/N (TπN (vq)) = 0. Then εTQ(AN ,AG/N ,AG) (see Lemma 6.3.8) isan isomorphism in the category LP.

Proof. Taking into account Lemmas 6.3.9 and 6.1.3 we see that we only need toshow that

εTQ(AN ,AG/N ,AG) (T (Q/G)) = T ((Q/N)/(G/N)) .

However, this follows easily from the definition of εTQ(AN ,AG/N ,AG).

The following corollary gives a structure theorem for Lie algebras.

Corollary 6.3.11. Let G be a Lie group, let N be a normal subgroup, and letK = G/N . Let g, n, and k be the Lie algebras of G, N , and K respectively. Choosean identification g ≡ k ⊕ n as linear spaces. Choose any connection AN on theprincipal bundle G with structure group N such that AN (gvq) = Adg AN (vq) forall g ∈ G, all vq ∈ TqG, and all q ∈ G. Then the Lie algebra bracket on g can bewritten in terms of the Lie algebra brackets on n and k and also ∇(AN ,V ) and BANas follows:

[κ1 ⊕ η1, κ2 ⊕ η2] = [κ1, κ2]⊕ [∇(AN ,V )]G/N,κ1η2 − [∇(AN ,V )]G/N,κ2η1

− [BAN ]G/N (κ1, κ2) + [η1, η2].

Proof. The proof is a straightforward application of Theorems 6.2.10, 6.3.7,and 6.3.10.

Now we shall generalize Theorem 6.3.7, Lemma 6.3.8, and Theorem 6.3.10 forarbitrary objects TQ⊕V rather than the particular case V = 0 considered in thoseresults. We will give precise statements, but we shall omit the proofs since they areentirely similar to the proofs of Theorem 6.3.7, Lemma 6.3.8, and Theorem 6.3.10.


Theorem 6.3.12. Let W = TQ ⊕ V be an object of LP and let G ×W → Wbe an action in the category LP of a Lie group G on W ; in particular, this actioninduces a principal bundle structure on Q with structure group G. Let N be anormal subgroup of G and let AN be a connection on the principal bundle Q withstructure group N having the property that for any g ∈ G and any vq ∈ TqQ wehave AN (gvq) = Adg AN (vq). Consider the action of G/N on T (Q/N)⊕ n⊕ (V/N)such that its restriction to T (Q/N)⊕ n coincides with the action defined in Theorem6.3.7 and its restriction to V/N is the quotient action of the action of G on V by N(see Lemma 5.2.1). Then this is an action in the category LP. The quotient bundleof T (Q/N)⊕ n⊕ (V/N) by the group G/N with respect to any connection AG/N onthe bundle Q/N with structure group G/N is an object of LP.

Lemma 6.3.13. Assume the conditions of Theorem 6.3.12 and, in addition,choose a connection AG on the principal bundle Q with structure group G. LetK = G/N . Consider the following vector bundle isomorphisms:

(a) the quotient isomorphism (see Lemma 6.3.6)

[αWAN ]G/N : (W/N)/(G/N)→ (T (Q/N)⊕ n⊕ (V/N)) /(G/N),

(b) the isomorphism (see Lemma 6.3.1)

iWG/N : W/G→ (W/N)/(G/N),

(c) the isomorphism (see Definition 6.2.11)

αWAG : W/G→ T (Q/G)⊕ g⊕ (V/G),

(d) the isomorphism (see Definition 6.2.11)

αT (Q/N)⊕n⊕(V/N)AG/N

: (T (Q/N)⊕ n⊕ (V/N)) /(G/N)

→ T ((Q/N)/(G/N))⊕ k⊕ (n⊕ (V/N)) /(G/N),

where k is the Lie algebra of K.

Then the corresponding linear isomorphisms

(a’)

[αWAN ]G/N∗ : Γ ((W/N)/(G/N))→ Γ ((T (Q/N)⊕ n⊕ (V/N)) /(G/N)) ,

(b’)iWG/N∗ : Γ(W/G)→ Γ ((W/N)/(G/N)) ,

(c’)αWAG∗ : Γ(W/G)→ Γ (T (Q/G)⊕ g⊕ (V/G)) ,


(d’)

αT (Q/N)⊕n⊕(V/N)AG/N∗ : Γ ((T (Q/N)⊕ n⊕ (V/N)) /(G/N))

→ Γ(T ((Q/N)/(G/N))⊕ k⊕ (n⊕ (V/N)) /(G/N)

)are Lie algebra isomorphisms. Consider the composition

(e)εW(AN ,AG/N ,AG) = α

T (Q/N)⊕n⊕(V/N)AG/N

[αWAN ]G/N iWG/N (αWAG)−1,

which is a vector bundle isomorphism from T (Q/G)⊕ g⊕ (V/G) onto

T ((Q/N)/(G/N))⊕ k⊕ (n⊕ (V/N)) /(G/N).

Then εW(AN ,AG/N ,AG) induces a Lie algebra isomorphism

(e’)

εW(AN ,AG/N ,AG)∗ : Γ((T (Q/G)⊕ g⊕ (V/G))

→ Γ(T ((Q/N)/(G/N))⊕ k⊕ n/(G/N)⊕ ((V/N)/(G/N))

).

Theorem 6.3.14. Assume the hypotheses of Theorem 6.3.12 and, in addition,choose the connections AN , AG/N and AG in such a way that the following propertyis satisfied: for all vq ∈ TQ and all q ∈ Q we have AG(vq) = 0 if and only ifAN (vq) = 0 and AG/N (TπN (vq)) = 0. Then εW(AN ,AG/N ,AG) is an isomorphism inthe category LP.

6.4 The Subcategory RT and Reduction byStages of Variational Principles on TQ.

Let T be the category of tangent bundles of manifolds and tangent lifts of maps,which is a subcategory of LP. The category RT is defined as the smallest subcat-egory of LP that contains T and is closed under the quotienting operation. Thus,as vector bundles, elements of RT are constructed by an inductive procedure of thetype V0 = TQ, Vi+1 = Vi/Gi, where for each i = 1, 2, . . ., Gi is a group acting on Vi.Using this and the fact that tangent bundles are always orientable manifolds we caneasily prove that objects of RT are orientable manifolds. This shows in particularthat RT does not coincide with LP since there are simple examples of objects ofLP that are nonorientable manifolds. For instance, let Q ≡ S1 and consider thevector bundle TQ⊕M , where M is the Mobius band; thus, as a manifold, TQ⊕Mis nonorientable. The Lie algebra structure on the fibers of M must be 0 becausethe fibers are 1-dimensional. The M -valued 2-form ω must be 0 because Q is 1-dimensional. Now choose a vector bundle metric on M . Then define the connection

6.4 The Subcategory RT and Reduction by Stages 95

∇ on the vector bundle M by the condition that a curve in M is horizontal if andonly if its distance to the 0 section is constant. Then we can check that, with thisstructure, TQ⊕M is an object of LP.

It is immediate that all the results of the previous sections involving genericobjects of LP are valid for generic objects of RT. A study of the local structureof objects of RT or LP and the study of categories bigger that LP, which appearin the study of nonholonomic systems, Poisson geometry, and other topics is beingplanned for future work. See Cendra, Marsden and Ratiu (2000) for work in thisdirection for nonholonomic systems. In this section, we shall present the theory ofLagrangian reduction by stages for the bundle TQ, where Q is a principal bundlewith structure group G having a normal subgroup N . This theory explains howHamilton’s variational principle corresponding to a given G-invariant, and henceN -invariant, Lagrangian L : TQ → R is reduced by stages, first under the actionof N and then under the action of G/N . The reduced bundles for these reducedvariational principles and corresponding Lagrange-Poincare operators are elementsof RT, as we saw in section §3.2, where the case of the first stage reduction wasstudied. A more general theory of reduction by stages of variational principles forany given object of LP including the reduction of the corresponding variationalprinciples is being planned for future work.

We shall begin with some remarks on the geometry of variations, using a notationslightly more precise than the one used in sections §2.1, §3.1, and §3.2. In thissection, we will often use, for short, the notation TQ⊕ V to denote an element ofRT, meaning that this bundle is isomorphic to a bundle of the type T (M/K)⊕ k,where M is a principal bundle with structure group G.

Let Q be a manifold. The tangent lift of a given curve q ∈ Ω(Q) is the curve`(q) in q ∈ Ω(TQ) defined by `(q)(t) = (q(t), q(t)) for all t; thus we obtain a map` : Ω(Q)→ Ω(TQ).

The space of allowed variations of a curve q is the tangent space ∆q ≡TqΩ(Q; q0, q1). Thus, elements of ∆q are variations δq such that δq(ti) = 0 fori = 1, 2, where, as usual, δq is the derivative of some deformation qλ(t), that is, forall t, we have

δq(t) =d

dλqλ(t)

∣∣∣∣λ=0

.

We have a canonical inclusion map TqΩ(Q)→ Ω(TQ). The space of lifted allowedvariations of a curve `(q) is the subspace ∆`

q of T`(q)Ω(TQ) defined by

∆`q = Tq` (TqΩ(Q; q0, q1)) ,

whereTq` : TqΩ(Q)→ T`(q)Ω(TQ)

is the tangent map of `. A generic element of ∆`q can therefore be written Tq` · δq,

for some variation δq of q such that δq(ti) = 0 for i = 1, 2. It is easy to prove thatthe restriction of Tq` to TqΩ(Q; q0, q1) ≡ ∆q is a linear bijective map onto ∆`

q.The following definition is inspired by Lemmas 3.1.4 and 3.1.7.


Definition 6.4.1. Let TQ ⊕ V be an object of RT and let τ : V → Q be theprojection. We denote by `Ω(Q) ⊕ Ω(V ) the set of all `(q) ⊕ v ∈ Ω(TQ ⊕ V ).Likewise `Ω(Q; q0) ⊕ Ω(V ; q0) is the subset of all `(q) ⊕ v ∈ Ω(TQ ⊕ V ) such thatq(t0) = q0 and `Ω(Q; q0, q1)⊕Ω(V ; q0, q1) is the subset of all `(q)⊕ v ∈ Ω(TQ⊕V )such that q(t0) = q0 and q(t1) = q1. Let v be a curve in V and let q = τv. Wewill sometimes think of the manifold Q as being identified with the vector bundlehaving base Q and 0-dimensional fiber. Thus if q = τv we can write v ≡ q ⊕ v (seesection §2.1) The space of allowed variations of q ⊕ v is the subspace ∆q⊕v ofTqΩ(Q)⊕ Ω(V ) of all curves of the type

δq ⊕ δv = δq ⊕ Dw

Dt+ [v, w]− ω(q)(q, δq),

where δq is a variation of q with δq(ti) = 0 for i = 1, 2, that is, δq ∈ ∆q, and w is acurve in V such that τ (w(t)) = q(t) for all t and w(ti) = 0 for i = 1, 2. The spaceof lifted allowed variations of a curve `(q)⊕ v ∈ Ω(TQ⊕ V ) is the subspace

∆`q⊕v := ∆`

q ⊕ Ω(V ) of T`(q)Ω(TQ)⊕ Ω(V )

of all variations of the type

Tq` · δq ⊕Dw

Dt+ [v, w]− ω(q)(q, δq),

where δq is a variation of q with δq(ti) = 0 for i = 1, 2, w is a curve in V such thatτ (w(t)) = q(t) for all t, and w(ti) = 0 for i = 1, 2.

Assume that the group G acts on the manifold Q. Then we have maps (seeLemma 5.2.1 and Lemma 2.4.2)

πG(TQ) : TQ→ TQ/G and αA : TQ/G→ T (Q/G)⊕ g.

Let `(q) ∈ Ω(TQ). Then we have

Ω (αA πG(TQ)) (`(q)) = `(x)⊕ ξ,

wherex(t) = [q]G(t) and ξ(t) = [q(t), A (q(t), q(t))]G

for all t.

Lemma 6.4.2. Assume the hypothesis of Lemma 2.4.2. Then the map

Ω (αA πG(TQ)) : Ω(TQ)→ Ω (T (Q/G)⊕ g)

restricted to `Ω(Q; q0) is injective. The image of this restriction is

`Ω (Q/G;x0)⊕ Ω (g;x0) .


Proof. Let x(t) ⊕ ξ(t) = `(x)(t) ⊕ ξ(t) be given such that x(t0) = x0. We mustshow that there is a unique curve q ∈ Ω(Q; q0) such that

Ω(αA πG(TQ))`(q) = `(x)⊕ ξ.

We can always write ξ(t) = [xhq0(t), ξ(t)]G. Let q(t) = g(t)xhq0(t) where g(t) is acurve in G. Then we can see that

Ω(αA πG(TQ))`(q) =(x(t)⊕ [xhq0(t), g(t)g−1(t)]G

).

Thus, g(t) must satisfy g(t) = ξ(t)g(t) for all t and g(t0) = e.

Corollary 6.4.3. The map

Ω (αA πG(TQ)) : Ω(TQ)→ Ω (T (Q/G)⊕ g)

restricted to `Ω(Q; q0, q1) is injective.

Lemma 6.4.4. Assume the hypothesis of Lemma 6.4.2. Then the restriction ofthe map TqΩ (αA πG(TQ)) to ∆`

q is a linear isomorphism onto ∆`x⊕ξ, where

x⊕ ξ = `(x)⊕ ξ = Ω(αA πG(TQ))`(q)

and ∆`x⊕ξ is the space of lifted allowed variations of `(x)⊕ ξ (see Definition 6.4.1).

Proof. Using Lemmas 3.1.4, 3.1.7, and 6.4.2 and considering that an elementof ∆`

q can be represented as the derivative of a deformation ` ((qλ(t)) of a curveq(t) ≡ q0(t) at λ = 0, it follows easily that

TqΩ (αA πG(TQ)) ∆`q ⊂ ∆`

x⊕ξ.

Bijectiveness of the restriction TqΩ (αA πG(TQ)) |∆`q can be proved using a re-

construction procedure as in the proof of Lemma 6.4.2. Namely, let `(xλ) ⊕ ξλ begiven a deformation of `(x)⊕ ξ, where ξλ(t) = [xhq0λ(t), ξλ(t)]G. This gives a uniquedeformation gλ(t) of g(t) satisfying gλ(t) = ξλ(t)gλ(t), which in turn gives a uniquedeformation qλ(t) = gλ(t)xhq0λ(t) of q(t). Therefore by differentiation with respectto λ at λ = 0 we obtain the inverse of the tangent map TqΩ (αA πG(TQ)).

Lemma 6.4.2 and Corollary 6.4.3 can be easily generalized for any object of LPinstead of TQ. However, a generalization of Lemma 6.4.4 involves a careful studyof the geometry of the space of variations and will be the purpose of future workon general Lagrangian reduction in the category LP. More precisely, we have

Lemma 6.4.5. Assume the hypothesis of Lemma 6.4.2 and, in addition, let W =TQ⊕ V be an element of LP. Then the map

Ω(αWA πG(W )

): Ω(W )→ Ω (T (Q/G)⊕ g⊕ (V/G))

restricted to `Ω(Q; q0)⊕ Ω(V ; q0) is injective. The image of this restriction is

`Ω(Q/G;x0)⊕ Ω(g;x0)⊕ Ω(V/G;x0).


Proof. For a given curve

x(t)⊕ ξ(t)⊕ [v]G(t) with x(t0) = x0

we find, using the curve x(t) ⊕ ξ(t), the curve q ∈ Ω(Q; q0) as in Lemma 6.4.2.Then using Lemma 5.2.1 we see that there is a unique curve v ∈ Ω(V ) such thatτv(t) = q(t), where τ : V → Q is the projection of V , and its class is [v]G(t) for allt.

We now state the generalization of Corollary 6.4.3.

Corollary 6.4.6. The map

Ω(αWA πG(W )

): Ω(W )→ Ω (T (Q/G)⊕ g⊕ (V/G))

restricted to ` (Ω(Q; q0, q1))⊕ Ω(V ) is injective.

Next we will see how the space of lifted allowed variations is transformed underreduction by stages.

Lemma 6.4.7. Assume the hypothesis of Theorem 6.3.14. Then we have

εW(AN ,AG/N ,AG) αWAG πG(W )

= αT (Q/N)⊕n⊕(V/N)AG/N

πG/N (T (Q/N)⊕ n⊕ (V/N)) αWAN πN (W ).

Proof. We can prove in a straightforward manner that

iWG/N πG(W ) = πG/N (W/N) πN (W )

[αWAN ]G/N πG/N (W/N) = πG/N (T (Q/N)⊕ n⊕ (V/N)) αWAN .

Using this and the definition of εW(AN ,AG/N ,AG) the lemma follows.

Theorem 6.4.8. Assume the hypothesis of Theorem 6.3.10. Then we have

TΩ(εTQ(AN ,AG/N ,AG)

) TΩ

(αTQAG πG(TQ)

)= TΩ

(αT (Q/N)⊕nAG/N

πG/N (T (Q/N)⊕ n)) TΩ

(αTQAN πN (TQ)

).

Let q ∈ Ω(Q) and let

Ω(αTQAG πG(TQ)

)(`(q)) = [q]G ⊕ ξ,

Ω(αTQAN πN (TQ)

)(`(q)) = [q]N ⊕ η,

Ω(αT (Q/N)⊕nAG/N

πG/N (T (Q/N)⊕ n))

(`([q]N ⊕ η)) = [[q]N ]G/N ⊕ κ⊕ [η]G/N .


Then

TΩ(αTQAG πG(TQ)

): ∆`

q → ∆`[q]G⊕ξ,

TΩ(αTQAN πN (TQ)

): ∆`

q → ∆`[q]N⊕η,

TΩ(αT (Q/N)⊕nAG/N

πG/N (T (Q/N)⊕ n))

: ∆`[q]N⊕η → ∆`

[[q]N ]G/N⊕κ⊕[η]G/N,


): ∆`

[q]G⊕ξ → ∆`[[q]N ]G/N⊕κ⊕[η]G/N

are linear isomorphisms.

Proof. The first equality in the statement of the theorem is a consequence ofLemma 6.4.7 for V = 0. The fact that

TΩ(αTQAG πG(TQ)

): ∆`

q → ∆`[q]G⊕ξ,

TΩ(αTQAN πN (TQ)

): ∆`

q → ∆`[q]N⊕η,


): ∆`

[q]G⊕ξ → ∆`[[q]N ]G/N⊕κ⊕[η]G/N

are linear isomorphisms is a consequence of Lemma 6.4.4 and Theorem 6.3.10. Thenit follows from the first equality in the statement of the theorem that also

TΩ(αT (Q/N)⊕nAG/N

πG/N (T (Q/N)⊕ n))

: ∆`[q]N⊕η → ∆`

[[q]N ]G/N⊕κ⊕[η]G/N

is a linear isomorphism.

The previous theorem says that, in the process of reducing the bundle TQ bystages, the spaces of lifted allowed variations can also be reduced by stages. Thisleads to reducing variational principles by stages, as we will see next.

First we need to generalize the notion of action defined by a Lagrangian. LetTQ ⊕ V be an object of LP, where τ : V → Q is the projection of V , and thestructure on V is given by ∇, ω, [ , ]. Let

L : TQ⊕ V → R

be a given Lagrangian, let v ∈ Ω(V ) be a given curve, and let q = τv. The actionof L at the curve v ∈ Ω(V ) is, by definition,

J(L)(q) =∫ t1

t0

L(q, q, v)dt.

We will also call this quantity the action of L at the curve `(q)⊕v of Ω(TQ⊕V )and denote it also by J(L) (`(q)).

The following definitions are obviously inspired by the results of §3.2. The Lagrange-Poincare operator is a bundle map

LP(L) : T (2)Q×Q/G 2V → T ∗Q⊕ V ∗


defined by

(LP)(L)(δq ⊕ δv) = Hor(LP)(L)δq + Ver(LP)(L)δv,

where the vertical and horizontal Lagrange-Poincare operators are given by

Ver(LP)(L)δv =(− D

Dt

∂L

∂v(q, q, v) + ad∗v

∂L

∂v(q, q, v)

)δv,

Hor(LP)(L)δq =(∂L

∂q(q, q, v)− D

Dt

∂L

∂q(q, q, v)

)δq − ∂L

∂v(q, q, v)ω(q)(q, δq).

Here we must choose an arbitrary affine connection ∇ on Q to make sense of thecovariant derivatives, in a similar way as we explained in §3.2 for the case V = g.For instance, in local coordinates, we can choose the Euclidean connection.

Now assume that there is an action of the group G on TQ ⊕ V in the categoryLP and that the Lagrangian L is invariant under the action. Then for any choiceof a connection AG on the principal bundle Q we can identify

(TQ⊕ V ) /G with T (Q/G)⊕ g⊕ (V/G)

via the isomorphism αTQ⊕VAGand we obtain an induced Lagrangian

L(G) : T (Q/G)⊕ g⊕ (V/G)→ R,

called the reduced Lagrangian (to make the notation consistent with the one usedin the particular case V = 0 considered in Theorem 3.3.4 we should write l ≡ L(G),in that particular case). If, furthermore, L(G) is invariant under the action of anothergroup K on the object

T (Q/G)⊕ g⊕ (V/G)

in the category LP, and we choose a connection AK on the principal bundle Q/Gwith structure group K, then we will denote the reduced Lagrangian simply byL(G,K) instead of

(L(G)

)(K). We have

L(G,K) : T ((Q/G)/K)⊕ k⊕ (g/K)⊕ (V/G)/K)→ R.

As we said before, in this work we will explain how to perform reduction bystages only for the case V = 0, leaving a more general theory for future work.

Assume the hypothesis of Theorem 6.4.8, let K = G/N , and let L : TQ→ R bea given Lagrangian. Then we have reduced Lagrangians

L(G) : T (Q/G)⊕ g→ R,

L(N) : T (Q/N)⊕ n→ R

andL(N,K) : T ((Q/N)/K)⊕ k⊕ (n/K)→ R.


For any given curve q ∈ Ω(Q) let

Ω(αTQAG πG(TQ)

)(`(q)) = x⊕ ξ,

Ω(αTQAN πN (TQ)

)(`(q)) = y ⊕ η,

Ω(αT (Q/N)⊕nAK

πK (T (Q/N)⊕ n))

(`(y ⊕ η)) = z ⊕ κ⊕ [η]K ,

as in Theorem 6.4.8. The next theorem is the main result of this section.

Theorem 6.4.9. Assume the situation explained above. Then the following con-ditions are equivalent:

(i) The curve q(t) is a critical point of the action functional∫ t1

t0

L(q, q)dt

with restrictions on the variations given by ∆`q.

(ii) The Euler–Lagrange equations

(EL)(L)(q) = 0

are satisfied.

(iii) The curve x(t)⊕ ξ(t) is a critical point of the action functional∫ t1

t0

L(G)(x(t), x(t), ξ(t)

)dt

with restrictions on the variations given by ∆`x⊕ξ.

(iv) The Lagrange-Poincare equations

LP(L(G))(x⊕ ξ) = 0

are satisfied.

(v) The curve y(t)⊕ η(t) is a critical point of the action functional∫ t1

t0

L(N) (y(t), y(t), η(t)) dt

with restrictions on the variations given by ∆`y⊕η.

(vi) The Lagrange-Poincare equations

LP(L(N))(y ⊕ η) = 0

are satisfied.


(vii) The curve z ⊕ κ⊕ [η]K is a critical point of the action functional∫ t1

t0

L(N,K) (z ⊕ κ⊕ [η]K) dt

with restrictions on the variations given by ∆z⊕κ⊕[η]K .

(viii) The Lagrange-Poincare equations

LP(L(N,K))(z ⊕ κ⊕ [η]K) = 0

are satisfied.

We will omit the proof of this theorem. It can be easily performed using anargument similar to the one used in the proof of Theorems 3.3.4 and 3.4.1 and alsousing Theorem 6.4.8.

Remark.

The main point of Theorem 6.4.9 is the equivalence with the last two statements(vii) and (viii), since the equivalence between the statements (i) to (vi) has beenalready established in Theorem 3.4.1. This shows that one can write the Euler–Lagrange equations for the Lagrangian

L(N,K) : T ((Q/N)/K)⊕ k⊕ (n/K)→ R

using the reduced structures given by the formulas of Theorem 6.2.10. More pre-cisely, the structure on the bundle n is the one given by ∇AN , BAN , [ , ]AN . Thenfrom Theorem 6.2.10 we obtain the formulas for the structure ∇, ω, [ , ] on thebundle k⊕ n, with ∇ ≡ ∇AN , ω ≡ BAN , [ , ] ≡ [ , ]AN .

All this can be generalized for several stages, that is, for the case where we havea chain

e ≡ N0 ⊂ N1 ⊂ N2 ⊂ N3 . . . ⊂ Nr ≡ G,

where for each i = 0, 1, 2, . . . , r−1, Ni is a normal subgroup of Ni+1. This togetherwith some applications will be the purpose of future work.

Page 103

7Further Examples

We have already seen one example of Lagrange-Poincare reduction, namely in thestudy of Wong’s equations. Now we turn to some examples that involve reductionby stages. The first one concerns the classical setting of semidirect products inwhich one is given a Lagrangian that is invariant under the action of a semidirectproduct, for example, as in underwater vehicle dynamics (see Leonard and Marsden(1997)). Here we show how this theory fits into the framework of the present paper.

Following this, we consider central extensions from the Lagrangian viewpoint, thesort of example that is well known in the Hamiltonian context of the KdV equation,as in Marsden, Misiolek, Perlmutter and Ratiu [1998, 2000].

The third example is that of the spacecraft with internal rotors. The group inthis case is just a direct product, but it is nonetheless interesting. See, for example,Bloch, Krishnaprasad, Marsden and Alvarez (1992).

The Lagrangian version of systems with a Lagrangian depending on parameters,in which semidirect products are, in a sense, created, is studied in the fourth exam-ple. These systems include, for instance, the classical heavy top. This sort of theorywas studied in Holm, Marsden and Ratiu (1998a) and in Cendra, Holm, Marsdenand Ratiu (1998).

7.1 Semidirect Products

In this subsection we show that the reduction of a system having a symmetry groupthat is a semidirect product of a Lie group G with a vector space V can be done intwo stages, reducing by the normal subgroup V first and by the group G second.

104 7. Further Examples

Let G be a Lie group and let ρ : G × V → V be a linear representation of Gon a vector space V . We will write, equivalently, ρ(g, v) ≡ ρgv ≡ ρ(g)v ≡ gv. LetS = GsV be the semidirect product. Thus, by definition,

(g1, v1)(g2, v2) = (g1g2, g1v2 + v1)

and (g, v)−1 = (g−1,−g−1v). The Lie algebra of S is s = gsV with the Lie bracketgiven by

[(ξ1, v1), (ξ2, v2)] = ([ξ1, , ξ2], ρ′(ξ1)v2 − ρ′(ξ2)v1) .

We will think of V as being a normal subgroup of S, as usual, via the inclusionmap V → GsV given by v 7→ (e, v). Next, we will perform the reduction ofTS = T (GsV ) in the category LP in two stages, first we reduce by V and thenby G ≡ (GsV )/V .

Stage 1.

We can identify in a natural way T (GsV ) = TG × V × V . Let us choose thetrivial connection AV on the principal bundle GsV with structure group V , thatis AV (g, g, v, v) = v. Then the curvature BAV = 0. Also, by definition V is Abelianand we can identify its Lie algebra v ≡ V , where the Lie bracket is 0. We willalso identify (GsV )/V ≡ G by [g, v]V ≡ g. The bundle v is a trivial bundle withbase (GsV )/V ≡ G and we can identify v ≡ G × V via [(g, v), ξ]V ≡ (g, ξ). Thecovariant derivative on v is given by

DAV

Dt[(g(t), v(t)), ξ(t)]V ≡

DAV

Dt(g(t), ξ(t)) = (g(t), ξ(t)).

We also have BAV = 0, and [ , ]v = 0. The reduced bundle that is obtained afterthe first stage of reduction is then

T ((GsV )/V )⊕ v ≡ TG× V

with the structure given above.

Stage 2.

The action of G on v commutes with the structure ∇ ≡ ∇AV , ω ≡ BAV = 0,[ , ] ≡ [ , ]v = 0. Now consider the principal bundle G over a point with structuregroup G and connection AG(g, g) = gg−1. We are in position to apply Theorems6.2.10, 6.3.10 and 6.4.9. Reduction by G of the bundle TG×V obtained in the firststage gives, using Theorem 6.2.10, a reduced bundle which is clearly isomorphic, inthe category LP, to the bundle g⊕ V whose base has dimension 0. It is clear thatωg = 0, ∇g = 0, because the base has dimension 0. Besides, we can check very easilythat the formula for [ , ]g gives the usual semidirect product Lie algebra structureon gsV . We remark that even if the bundle obtained in the first stage depends onthe connection A, the bundle obtained in the second stage does not depend on theconnection A. This is, of course, also a direct consequence of Theorem 6.3.10.

7.2 Central Extensions 105

7.2 Central Extensions

We now study the particular case of R or S1-group extensions. This already in-cludes the Bott–Virasoro central extension of Diff(S1) as an interesting infinitedimensional example. More general central extensions by Abelian Lie groups, rela-tive to an arbitrary action (see, for instance, De Azcarraga and Izquierdo (1995)),can also be dealt with using the methods of this paper. For instance, the exam-ples studied in Marsden, Misiolek, Perlmutter and Ratiu [1998, 2000], such as theHeisenberg group and the Bott–Virasoro group treated there from a Hamiltonianpoint of view, can also be approached using our Lagrangian techniques.

Let G be a Lie group and consider central extensions of G of the type G = G×Ror G = G× S1, where the composition law is given by

(g, α)(h, β) = (gh, α+ β + Σ(g, h)) .

If we deal with G× S1, the sum in second component is understood to be modulo2π. Here, Σ : G×G→ R is a group 2-cocycle relative to the trivial action of G onR and hence Σ satisfies the cocycle identity

Σ(g, h) + Σ(gh, k) = Σ(g, hk) + Σ(h, k).

As is well known, the second group cohomology of G with values in an Abelian groupclassifies the extensions of G by this Abelian group. Therefore, in the definition ofthe composition law for G we can modify Σ by the addition of a 2-coboundarysuch that Σ(e, e) = 0. The cocycle identity implies that Σ(h, e) = Σ(e, h) = Σ(e, e)and Σ(h, h−1) = Σ(h−1, h) for all h ∈ G. It can be directly checked that thecocycle identity and the condition Σ(e, e) = 0, are necessary and sufficient for thecomposition law defined above to satisfy the group axioms. It is easily verified thatthe neutral element of G is (e, 0), where e is the neutral element of G. The inverseof (g, α) is given by

(g, α)−1 =(g−1,−α− Σ(g, g−1)

).

An element of the tangent space T(g,α)G is often denoted by (g, α, g, α) or (g, α).The Lie algebra g is the space of all tangent vectors at (e, 0), that is, vectors(e, 0, ξ, a), where ξ is an element of the Lie algebra g. With these basic formulas inplace we can calculate the Lie bracket in g which is given by

[(ξ, a), (η, b)] = ([ξ, η], ∂1∂2Σ(e, e) · η · ξ − ∂2∂1Σ(e, e) · η · ξ) .

Here we define

∂1∂2Σ(e, e) · η · ξ =∂

∂t

∣∣∣∣t=0

∂

∂s

∣∣∣∣s=0

Σ(exptξ, expsη)

and

∂2∂1Σ(e, e) · η · ξ =∂

∂t

∣∣∣∣t=0

∂

∂s

∣∣∣∣s=0

Σ(expsη, exptξ).


We will define the Lie algebra 2-cocycle σ by

σ(ξ, η) = ∂1∂2Σ(e, e) · η · ξ − ∂2∂1Σ(e, e) · η · ξ

and then the Lie bracket is [(ξ, a), (η, b)] = ([ξ, η], σ(ξ, η)).The set N = (e, α) | α ∈ R is a normal subgroup of G isomorphic to R or

S1. Moreover, N is contained in the center of G. We can therefore consider G as aprincipal bundle with structure group N . Now consider any metric on g, for instanceone given by the simple formula

〈(ξ, a), (η, b)〉 = 〈ξ, η〉+ ab,

where (by a slight abuse of notation), 〈ξ, η〉 is a given inner product on g. Thisgenerates a left invariant metric on G in the following way. Let (gi, αi) ∈ T(g,α)G,i = 1, 2. Then define

〈(g1, α1), (g2, α2)〉 =⟨g−1g1, g

−1g2

⟩+(α1 + ∂2Σ(g−1, g) · g1

) (α2 + ∂2Σ(g−1, g) · g2

).

Now we can proceed to perform the construction of the mechanical connection, thatis, the connection whose horizontal spaces are the orthogonal complements of thevertical spaces of the bundle G with structure group N . Then the conclusion of6.3.5 holds, that is, the corresponding connection 1-form given by

AN (g, α) = α+ ∂2Σ(g−1, g) · g.

is G-equivariant and not just N -equivariant.Now we apply the reduction by stages process given in Theorems 6.3.14 and

6.2.10. The first stage is reduction by N and we obtain the reduced bundle TG⊕ n,since G/N ≡ G. We obtain a structure ∇ ≡ ∇AN , B ≡ BAN , and [ , ] on thebundle n. Then, by applying Theorems 6.3.14 and 6.2.10 we see that we obtain anequivalent expression for the Lie bracket on g ≡ g⊕ [n]G. We can easily check that,in the expression of the Lie bracket of Theorem 6.2.10, the terms [∇(A,V )]Gξ1 [v2]G,[∇(A,V )]Gξ2 [v1]G, as well as the term [[v1]G, [v2]G] are all 0. Thus, in particular, thecocycle σ is minus the reduced curvature.

7.3 Rigid Body with Rotors

We will use the description of the rigid body with rotors given in Marsden andScheurle (1993b). The configuration space for a rigid body with three rotors alignedwith, say, the principal axes, is

Q = SO(3)× S1 × S1 × S1

with elements denoted (R, θ1, θ2, θ3), where the angles are relative to the carrier.The group structure on Q is the direct product structure, so, in particular, N =

7.4 Systems Depending on a Parameter 107

SO(3) is a normal subgroup. The body angular velocity is Ω = R−1R. We alsodenote

Ω = ˙(θ1, θ2, θ3).

We now think of Q as a principal bundle with structure group N and choose themechanical connection AN given by

AN (R, θ1, θ2, θ3) = R−1R+ (I +K)−1KΩ.

The first stage of reduction is reduction by N which gives the reduced bundle

T (S1 × S1 × S1)⊕ n ≡ T (S1 × S1 × S1)× n.

The second stage is reduction by the group S1 × S1 × S1 ≡ Q/N . We can easilysee that the reduced bundle is

R× R× R× n/S1 × S1 × S1 ≡ R× R× R× n.

We can also easily see that, in our case, several terms of the expression of the Liebracket given in Theorem 6.2.10 vanish; more precisely, [ξ1, ξ2] = 0, [∇(A,V )]Gξ1 [v2]G =0, [∇(A,V )]Gξ2 [v1]G = 0, and [ω]G(ξ1, ξ2) = 0. We can also easily check that theterm [[v1]G, [v2]G] is, in our case, simply [Ω1,Ω2]. Thus the Lie algebra structure onR × R × R × n obtained by reduction by stages coincides with the direct productLie algebra, as expected, according to Theorems 6.3.14 and 6.2.10.

7.4 Systems Depending on a Parameter

In this subsection, we consider Lagrangian systems depending on a parameter andshow how they fit into the framework developed in this paper. In the process weshall recover the Euler-Poincare equation in Holm, Marsden and Ratiu (1998a); seeequation (7.4.1) below.

Recall from Marsden, Ratiu, and Weinstein (1984a) and Marsden, Misiolek, Perl-mutter and Ratiu (1998) that on the Hamiltonian side of this same problem, thesemidirect product GsV appears as a symmetry group of an enlarged system andsymplectic reduction by stages is relevant. The situation on the Lagrangian side ofthe same problem is somewhat different.

The fundamental difference between this subsection and §7.1 is that in §7.1, oneimagines having a system whose given symmetry group is a semidirect productfrom the outset, as, for example, the Euclidean group is the symmetry group ofan underwater vehicle. Here, on the other hand, the Lagrangian has a symmetrygroup G, but in a way that includes the dependence on a parameter a ∈ V ∗, whereV is a representation space for G. The goal is then to show that the more generalEuler–Poincare equations referred to above, can be obtained by using Lagrangianreduction with respect to the action of G.

There are interesting links between the set up described below and the topic ofClebsch variables and Lin constraints; these are described in Cendra and Marsden


(1987). We also refer to Cendra, Holm, Marsden and Ratiu (1998) and Cendra,Holm, Hoyle and Marsden (1998) for interesting connections with degenerate Leg-endre transformations.

Consider a Lagrangian

L : T (G×Q)× V ∗ → R,

where G is a group, Q is a manifold, and V ∗ is the dual of the vector space V . Thevalue of L at the point (g, q, g, q, a0) ∈ T (G×Q)×V ∗ will be denoted L(g, q, g, q, a0),as usual, and we will think of a0 as being a parameter that remains fixed along theevolution of the system. Assume that there is a linear action ρ : G×V → V of G onV , so there is also an induced action ρ∗ : G×V ∗ → V ∗ such that 〈ga0, gb0〉 = 〈a0, b0〉for all a0 ∈ V ∗, all b0 ∈ V , and all g ∈ G, where, as usual, we write ga0 = ρ(g, a0)and gb0 = ρ∗(g, b0). We will often write 〈a0, b0〉 = 〈b0, a0〉, due to the identificationV ∗∗ ≡ V . Assume that L has the following invariance property:

L(hg, q, hg, q, ha0) = L(g, q, g, q, a0),

for all a0 ∈ V ∗, all q ∈ Q, and all h, g ∈ G. Let

L(e, q, ξ, q, a) = l(ξ, q, q, a),

for all ξ ∈ g, all q ∈ Q, and all a ∈ V ∗. Then the invariance property implies

L(g, q, g, q, a0) = l(ξ, q, q, a),

for all g ∈ G, all q ∈ Q, and all a ∈ V ∗, where ξ = g−1g and a = g−1a0.By direct calculation we can show that the following conditions are equivalent:

(i) The curve (g(t), q(t), a0) is a critical point of the action∫ t1

t0

L(g, q, g, q, a0)dt,

with restrictions on variations given by δg(ti) = 0 for i = 0, 1, δq(ti) = 0 fori = 0, 1, and δa0 = 0.

(ii) The curve (ξ(t), q(t), a(t)), where a(t) = g−1(t)a0 for all t and ξ(t) = g−1(t)g,is a critical point of the action∫ t1

t0

l(ξ, q, q, a)dt,

with restrictions on variations given by

δξ = η + [ξ, η],

where η is any curve in g such that η(ti) = 0 for i = 0, 1,

δq(ti) = 0,


for i = 0, 1,δa = −ηa,

anda+ ξa = 0,

for all t. We remark that this last condition comes from the condition a0 = 0together with a0 = ga.

A direct application of (ii) leads to the equations

− d

dt

∂l

∂ξ+ ad∗ξ

∂l

∂ξ+∂l

∂a a = 0 (7.4.1)

∂l

∂q− d

dt

∂l

∂q= 0, (7.4.2)

where for all η ∈ g, all a ∈ V ∗, and all b ∈ V we have, by definition,

(b a)(η) = −〈ηa, b〉 .

The equation (7.4.1) is called the Euler–Poincare equation (see Holm, Mars-den and Ratiu (1998a)). This equation together with the Euler–Lagrange equation(7.4.2) in the variable (q, q) and the equation a + ξa = 0 form the complete set ofequations of the system in terms of the variables (ξ, q, a).

Now we shall recast conditions (i) and (ii) into an equivalent form. The idea isto introduce the condition that a0 is a constant of the motion, that is, a0 = 0, viaa Lagrange multiplier. Thus, let us define the new Lagrangian

L : T (G×Q× V ∗ × V )→ R

byL(g, q, a0, b0, g, q, a0, b0) = L(g, q, g, q) + 〈a0, b0〉 .

Now we observe that G×Q×V ∗×V is a principal bundle with structure group Gacting as before, that is, h(g, q, a0.b0) = (hg, q, ha0, hb0). Moreover, G×Q×V ∗×Vis isomorphic, as a principal bundle, to the trivial bundle G × Q × V ∗ × V wherethe action of G is given by

h · (g, q, a, b) = (hg, q, a, b),

for all h, g ∈ G, all a ∈ V ∗, and all b ∈ V . More precisely, we have the isomorphism

ψ : G×Q× V ∗ × V → G×Q× V ∗ × V

given byψ(g, q, a0, b0) = (g, q, g−1a0, g

−1b0) ≡ (g, q, a, b).

We can check that

ψ (h(g, q, a0, b0)) = h · ψ(g, q, a0, b0) ≡ h · (g, q, a, b)


for all h, g ∈ G, all a0 ∈ V ∗, and all b0 ∈ V . We can easily check that the compositionL Tψ−1 =: LV is given by

LV (g, q, a, b, g, q, a, b) = L(g, q, g, q, a) +⟨a+ g−1ga, b

⟩.

We will use the trivial bundle, where the action of G is given by the · operation,from now on. Thus, the base is

(G×Q× V ∗ × V )/G ≡ Q× V ∗ × V

in a natural way. The Lagrangian on this bundle is LV . Using techniques as inCendra and Marsden (1987), which gives a version of the Lagrange multiplier theo-rem, we can show that conditions (i) and (ii) are equivalent to any of the followingconditions

(iii) The curve (g(t), q(t), a0, b0) is a critical point of the action∫ t1

t0

L(g, q, a0, b0, g, q, a0, b0)dt

with restrictions on the variations given by

δg(ti) = 0, δq(ti) = 0, δa0(ti) = 0, and δb0(ti) = 0

for i = 0, 1.

(iv) The curve (g(t), q(t), a(t), b(t)) is a critical point of the action∫ t1

t0

LV (g, q, a, b, g, q, a, b)dt

with restriction on the variations given by

δg(ti) = 0, δq(ti) = 0, δa(ti) = 0, and δb(ti) = 0

for i = 0, 1.

However, this time we do not want to use that version of the Lagrange multipliertheorem. Instead, we want to use directly Theorem 3.3.4 to obtain a reduced system,starting with the Lagrangian LV , equivalent to the system

− d

dt

∂l

∂ξ+ ad∗ξ

∂l

∂ξ+∂l

∂a a = 0

∂l

∂q− d

dt

∂l

∂q= 0

a+ ξa = 0

(7.4.3)

obtained before.


We need to calculate the reduced bundle T (Q×V ∗×V )⊕ g. We can easily verifythat the bundle g equals Q× V ∗ × V × g. The Lie algebra structure on g is givenby

[(q, a, b, ξ1), (q, a, b, ξ2)] = (q, a, b, [ξ1, ξ2]).

Now let us choose the trivial principal connection A on G × Q × V ∗ × V , that is,the connection given by

A(g, q, a, b, g, q, a, b) = gg−1.

Using Lemma 2.3.4 we can see that the covariant derivative along a curve in g isgiven by

D

Dt(q(t), a(t), b(t), ξ(t)) =

(q(t), a(t), b(t), ξ(t)

).

Therefore the connection ∇A on g is given by

∇A(q,a,b,q,a,b)

(q, a, b, ξ) =(q, a, b,

∂ξ

∂qq∂ξ

∂aa+

∂ξ

∂bb

),

for any section (q, a, b) 7→ (q, a, b, ξ(q, a, b)) of g and any tangent vector

(q, a, b, q, a, b)

at the point (q, a, b) of the base Q×V ∗×V of g. Since the curvature of A is BA = 0we have BA = 0. A generic element of the bundle

T (Q× V ∗ × V )⊕ g

will be written (q, a, b, q, a, b, ξ). Let lV be the reduced Lagrangian

lV : T (Q× V ∗ × V )⊕ g→ R.

Thus, lV (q, a, b, q, a, b, ξ) is a function of the independent variables

(q, a, b, q, a, b, ξ).

More precisely, we can easily see that

lV (q, a, b, q, a, b, ξ) = l(q, ξ, q, a) + 〈a+ ξa, b〉 ,

where, as before,l(ξ, q, q, a) = L(e, q, ξ, q, a).

We can read off directly from the expression of the Lagrangian lV above that thecondition a + ξa = 0 has been imposed with the Lagrange multiplier b. We canwrite (see Cendra and Marsden (1987))

lV (q, a, b, q, a, b, ξ) = l(q, ξ, q, a) + J(a, b)(ξ) + θ0(a, b)(a, b),


where J : V ∗ × V → g∗ is the momentum map of the lift of the action ρ tothe cotangent bundle V ∗ × V ≡ V ∗ × V ∗∗, and θ0 is the canonical 1-form onV ∗ × V ≡ V ∗ × V ∗∗. Using Theorem 3.3.4 we can obtain the vertical and also thehorizontal Lagrange–Poincare equations. The vertical Lagrange–Poincare equationis

− d

dt

(∂l

∂ξ(ξ, q, q, a) + J(a, b)

)+ ad∗ξ

(∂l

∂ξ(ξ, q, q, a) + J(a, b)

)= 0.

The horizontal Lagrange-Poincare equation is

∂l

∂q(ξ, q, q, a)− d

dt

∂l

∂q(ξ, q, q, a) = 0

∂l

∂a(ξ, q, q, a) +

∂J(a, b)(ξ)∂a

− db

dt= 0

a+ ξa = 0.

Using the property 〈ξa, b〉+ 〈a, ξb〉 = 0, we can see that

∂J(a, b)(ξ)∂a

= −ξb.

The last two horizontal equations can therefore be rewritten as

b = −ξb+∂l

∂aa = −ξa.

It is also clear thatdJ(a, b)dt

= dJ(a, b)(a, b).

Thus we obtaindJ(a, b)dt

= dJ(a, b)(−ξa,−ξb+

∂l

∂a

).

Using directly the formula J(a, b)(ν) = 〈νa, b〉 = −〈a, νb〉 we can prove in astraightforward manner that

dJ(a, b)(−ξa,−ξb+

∂l

∂a

)= ad∗ξ J(a, b)− ∂l

∂a a.

Using this, together with the vertical Lagrange–Poincare equation, we obtain ateach point (ξ, q, q, a),

− d

dt

(∂l

∂ξ

)+∂l

∂a a+ ad∗ξ

(∂l

∂ξ

)= 0,

which is the Euler–Poincare equation. We can easily conclude from all this that anysolution to the reduced system of equations is a solution of the system (7.4.3).


We can also prove the converse, namely, starting with a solution (q(t), a(t), ξ(t))of the system (7.4.3) we can see that the curve b(t) = g(t)b0, for any given b0 ∈ V ,is such that (q(t), a(t), b(t), ξ(t)) is a solution of the system formed by the reducedvertical and horizontal Euler–Poincare equations. This is done by simply reversingthe previous procedure.


Page 115

8The Category LP∗ and Poisson Geometry

In this section we will define a new category LP∗. The objects of LP∗ are the dualbundles of the objects in LP and they carry a Poisson structure which is dual to theLie bracket structure on sections of objects of the category LP. Cotangent bundlesare important examples of objects of LP∗. The parallelism between Lagrangianmechanics and Poisson mechanics is a consequence of the Legendre transformation.Interesting aspects of this parallelism can be viewed under the light of the cate-gorical duality. For instance, reduction in the category LP∗ is dual to reduction inthe category LP. This includes cotangent bundle reduction as a special case (seeMontgomery, Marsden and Ratiu (1984) and Montgomery (1986)). Thus, LP∗ isan interesting class of Poisson manifolds which is stable under reduction.

In this section we establish the basic link between Lagrangian and Poisson me-chanics given by this duality and show how reduction in the category LP∗, whichis a subcategory of the category of Poisson manifolds and Poisson maps, can beviewed as being dual to reduction in the category LP. A more complete study, in-cluding a precise description of the symplectic leaves of objects of LP∗ and severalother related topics will be the purpose of a future work.

8.1 The Poisson Bracket on Duals of Objects ofLP

We shall begin with some notation and definitions. Let W = TQ⊕V be an object ofLP and let π : W → Q, pQ : W → TQ and pV : W → V be the natural projections.

116 8. The Category LP∗ and Poisson Geometry

Let W ∗ = (TQ⊕ V )∗ ≡ T ∗Q⊕ V ∗ be the dual of W and let

π : W ∗ → Q, pQ : W ∗ → T ∗Q, and pV : W ∗ → V ∗

be the naturally induced projections. A section w ∈ Γ(W ) will be sometimes writtenmore explicitly as w = X ⊕ v or w = q ⊕ v. Likewise, a section µ ∈ Γ(W ∗) will besometimes written µ = γ ⊕ ν. We have well defined maps

Γ(pQ) : Γ(W )→ Γ(TQ) and Γ(pV ) : Γ(W )→ Γ(V )

and also

Γ(pQ) : Γ(W ∗)→ Γ(T ∗Q) and Γ(pV ) : Γ(W ∗)→ Γ(V ∗).

We will often denote

Γ(pQ)(w) ≡ pQw and Γ(pV )(w) ≡ pV w

and alsoΓ(pQ)µ ≡ pQµ and Γ(pV )(µ) ≡ pV µ.

Thus, if we write a section w ∈ Γ(W ) as w = X ⊕ v and a section µ ∈ Γ(W ∗) asµ = γ ⊕ ν, we have

Γ(pQ)(w) ≡ pQw = X and Γ(pV )(w) ≡ pV w = v

and alsoΓ(pQ)µ ≡ pQµ = γ and Γ(pV )(µ) ≡ pV µ = ν.

Definition 8.1.1. For each w ∈ Γ(W ) we define the function P (w) ∈ C∞(W ∗)by

P (w)(µ) = µ(w)

for all µ ∈ W ∗. In addition, for each f ∈ C∞(Q) we have the function f π ∈C∞(W ∗).

Define the space A(W ∗) ⊂ C∞(W ∗) to be the vector space generated by the setof all functions f π with f ∈ C∞(Q), together with the set of all functions thatare linear along the fibers of C∞(W ∗), that is, the functions of the type P (w) withw ∈ Γ(W ).

Lemma 8.1.2. Let wi ∈ Γ(W ) and fi ∈ C∞(Q), for i = 1, 2. Define We willsometimes write, for given f ∈ C∞(Q) and given w ∈ Γ(W ), f ≡ f π andw[f ] ≡ (Γ(pQ)(w)) [f ] π, for short.

Then , extends to a uniquely determined skew-symmetric bilinear map

, : A(W ∗)×A(W ∗)→ A(W ∗).

The operation , satisfies the Jacobi identity, that is,

f1, f2, f3 = f1, f2, f3+ f2, f1, f3

for all fi ∈ A(W ∗), i = 1, 2, 3.

8.1 The Poisson Bracket on Duals of Objects of LP 117

Proof. Every element of A(W ∗) can be written f π + P (w) for some uniquelydetermined f ∈ C∞(Q) and w ∈ Γ(W ). Using this we can prove in a straightfor-ward manner that , extends to a uniquely determined skew-symmetric bilinearform on A(W ∗). The verification that , satisfies the Jacobi identity is also astraightforward calculation.

Theorem 8.1.3. The operation , on A(W ∗) defined in the previous lemma canbe uniquely extended to a Poisson bracket

, : C∞(W ∗)× C∞(W ∗)→ C∞(W ∗).

Proof. The proof will be divided into several steps.

Step 1:

The definition of the bracket.

Let fi ∈ C∞(W ∗), i = 1, 2, and σ0 ∈W ∗ be given and denote π(σ0) = q0. We aregoing to define f1, f2(σ0). Consider any section σ ∈ Γ(W ∗) such that σ(q0) = σ0.Then for each q ∈ Q we have the Taylor expansion, for i = 1, 2,

fi(µ) = fi (σ(q)) +∂fi∂µ

(σ(q)) (µ− σ(q)) + εi (σ(q), µ) · (µ− σ(q)) ,

where for each q ∈ Q we have µ ∈W ∗q ,

∂fi∂µ

(σ(q))

is the fiber derivative of fi along W ∗q evaluated at σ(q), and εi (σ(q), µ) → 0 asµ→ σ(q). Define wi ∈ Γ(W ) and ai ∈ C∞(Q) for i = 1, 2, by

wi(q) =∂fi∂µ

(σ(q))

andai(q) = fi (σ(q))− ∂fi

∂µ(σ(q))σ(q)

for all q ∈ Q. Thus, for i = 1, 2, ai+P (wi) is the affine approximation of fi along thefibers of W ∗. To emphasize the dependence on σ we will sometimes write aσi ≡ ai,wσi ≡ wi, and fσi ≡ aσi + P (wσi ), for i = 1, 2. The functions fσi are elements ofA(W ∗). Then we can define, for each choice of σ satisfying σ0 = σ(q0),

f1, f2σ,σ0(µ) = fσ1 , fσ2 (µ),

where the bracket on the right hand side is the bracket in A(W ∗) defined before.We want to show that

f1, f2σ,σ0(σ0)

only depends on σ0 and not on σ. To do this, we need first the following technicalstatement.


Step 2:

If dfi(σ0) = 0 for i = 1 or i = 2, then f1, f2σ,σ0(σ0) = 0.

We work in a local bundle chart of W = TQ ⊕ V whose restriction to TQis a natural tangent bundle chart. This can be done by first choosing a vectorbundle chart of V which induces a chart of Q and hence has a naturally associatedtangent bundle chart. The same bundle chart of W induces a bundle chart onW ∗ = T ∗Q ⊕ V ∗. An element of W will be represented in this local chart by(q, w) ≡ (q, q ⊕ v) and an element of W ∗ will be represented in the correspondinglocal chart by (q, µ) ≡ (q, p⊕ν). Thus we will write, with a slight abuse of notation,

σ0 = (q0, µ0) = (q0, p0 ⊕ ν0),

σ(q) = (q, µ(q)) = (q, p(q)⊕ ν(q)),

andwσi (q) ≡ (q, wσi (q)) ≡ (q, qσi (q)⊕ vσi (q)).

A straightforward calculation shows that, for i = 1, 2, we have dfi(σ0) = dfσi (σ0).It is also easy to see that dfσi (σ0) = 0 for i = 1 or i = 2 if and only if the followingequalities hold, for i = 1 or i = 2, respectively :

∂aσi∂q

(q0) + µ0∂wσi∂q

(q0) = 0

wσi (q0) = 0

or, equivalently, for i = 1, 2,

∂aσi∂q

(q0) + p0∂qσi∂q

(q0) + ν0∂vσi∂q

(q0) = 0

qσi (q0) = 0vσi (q0) = 0.

This is valid for any chart as before. From now on we will assume that the previousconditions are satisfied for i = 1. The case i = 2 can be established in an entirelyanalogous way. By definition we have

f1, f2σ,σ0(σ0) = fσ1 , fσ2 (σ0)= aσ1 + P (wσ1 ), aσ2 + P (wσ2 )(σ0)= wσ2 (aσ1 )(q0)− wσ1 (aσ2 )(q0)− P ([wσ1 , w

σ2 ]) (σ0).

Since wσ1 (q0) = 0 we have wσ1 (q0)(aσ2 )(q0) = 0. On the other hand, by definition, wehave

wσ2 (q0)(aσ1 )(q0) = (pQwσ2 )(aσ1 )(q0)

=∂aσ1∂q

(q0)(pQwσ2 )(q0)

=∂aσ1∂q

(q0)qσ2 (q0).

8.1 The Poisson Bracket on Duals of Objects of LP 119

By Definition 6.1.1 the Lie bracket in the category LP is given by

[wσ1 , wσ2 ] = [qσ1 ⊕ vσ1 , qσ2 ⊕ vσ2 ]

= [qσ1 , qσ2 ]⊕∇qσ1 v

σ2 −∇qσ2 v

σ1 − ω(qσ1 , q

σ2 ) + [vσ1 , v

σ2 ].

Since qσ1 (q0) = 0 and vσ1 (q0) = 0 we have

[wσ1 , wσ2 ](q0) = [qσ1 , q

σ2 ](q0)⊕−∇qσ2 v

σ1 (q0).

We can easily show that

[qσ1 , qσ2 ](q0) = −∂q

σ1

∂q(q0)qσ2 (q0).

Now we choose the chart on W in such a way that the Christoffel symbols of theconnection∇ on V vanish at q0, which we can assume without any loss of generality.Then we obtain

∇qσ2 vσ1 (q0) =

∂vσ1∂q

(q0)qσ2 (q0).

Therefore, using the previous equalities, we can conclude that

fσ1 , fσ2 (σ0) =∂aσ1∂q

(q0)qσ2 (q0) + p0∂qσ1∂q

(q0)qσ2 (q0) + ν0∂vσ1∂q

(q0)qσ2 (q0)

=(∂aσ1∂q

(q0) + p0∂qσ1∂q

(q0) + ν0∂vσ1∂q

(q0))qσ2 (q0)

= 0.

We have therefore proved the aforementioned property, namely dfi(σ0) = 0 fori = 1 or i = 2 implies that f1, f2σ,σ0(σ0) = 0.

Step 3:

The bracket is independent of σ.

Using the above property, it follows by a standard argument that for given fi, gi ∈C∞(W ∗), i = 1, 2, and σ ∈ Γ(W ∗) satisfying dfi(σ0) = dgi(σ0) and σ(q0) = σ0, wehave

f1, f2σ,σ0(σ0) = g1, g2σ,σ0(σ0).

We can easily check that if fi ∈ A(W ∗) , for i = 1, 2, then for any section σ ∈ Γ(W ∗)and any choice of σ0 such that σ(q0) = σ0 we have fi = fσi , for i = 1, 2, and also

f1, f2(σ0) = f1, f2σ,σ0(σ0).

On the other hand, we can show, using the Taylor expansion, that for any choiceof a section σ′ ∈ Γ(W ∗) satisfying σ′(q0) = σ0 and any f ∈ C∞(W ∗) we have


dfσ′(σ0) = df(σ0) = dfσ(σ0). Using all this and the previous step, we can see

that

f1, f2σ,σ0(σ0) = fσ1 , fσ2 σ,σ0(σ0)

= fσ′

1 , fσ′

2 σ,σ0(σ0) = fσ′

1 , fσ′

2 σ′,σ0(σ0) = f1, f2σ

′,σ0(σ0).

So far, we have proved that the bracket f1, f2σ,σ0(σ0) is well defined, that is, itdoes not depend on the choice of the section σ ∈ Γ(W ∗) as long as σ(q0) = σ0.Thus, we shall denote this bracket from now on only by f1, f2(σ0).

It is clear that the well defined operation f1, f2(σ0) is bilinear and skew sym-metric. We need to show that it satisfies the Jacobi identity.

Step 4:

Verification of the Jacobi identity.Let fi ∈ C∞(W ∗), i = 1, 2, 3, and σ0 ∈W ∗. We need to verify that

f1, f2, f3(σ0) = f1, f2, f3(σ0) + f2, f1, f3(σ0).

To do this, choose σ ∈ Γ(W ∗) such that σ(q0) = σ0. Using what we have proven sofar we can see that the question reduces itself to showing that

fσ1 , fσ2 , fσ3 (σ0) = fσ1 , fσ2 , fσ3 (σ0) + fσ2 , fσ1 , fσ3 (σ0).

However, this identity holds by Lemma 8.1.2 since fσi ∈ A(W ∗), for i = 1, 2, 3.The last axiom for a Poisson bracket is the Leibnitz identity. We shall verify it

below.

Step 5:

Verification of the Leibnitz identity.

Let fi ∈ C∞(W ∗), i = 1, 2, 3, and σ0 ∈W ∗. We must prove that

f1, f2f3(σ0) = f1, f2(σ0)f3(σ0) + f2(σ0)f1, f3(σ0).

As we have seen before, we can assume without loss of generality that fi = fσi ∈A(W ∗), for i = 1, 2, 3, so we can write fi = ai + P (wi), for ai ∈ C∞(Q) andwi ∈ Γ(W ). Using the bilinearity and skew-symmetry of , , we can easily see thatit is sufficient to consider the following six particular cases:

f1 ≡ a1, f2 ≡ a2, f3 ≡ a3;f1 ≡ a1, f2 ≡ a2, f3 ≡ P (w3);f1 ≡ a1, f2 ≡ P (w2), f3 ≡ P (w3);f1 ≡ P (w1), f2 ≡ a2, f3 ≡ a3;f1 ≡ P (w1), f2 ≡ a2, f3 ≡ P (w3);f1 ≡ P (w1), f2 ≡ P (w2), f3 ≡ P (w3).

8.2 Poisson Reduction in the Category LP∗ 121

The first, second, fourth, and fifth cases can be established using directly the def-inition of the bracket on A(W ∗) and the fact that for any a ∈ C∞(Q) and anyw ∈ Γ(W ) we have aP (w) = P (aw).

To prove the third case, we first write the Taylor expansion of P (w2) and P (w3),namely

P (wi)(q, µ) = σ(q)(wi)(q) + (µ− σ(q)) (wi)(q)

for i = 2, 3. Then, since

d ((µ− σ(q)) (w2)(q) · (µ− σ(q)) (w3)(q)) = 0

at (q, µ) = σ0, we see that

a1, P (w2)P (w3)(σ0) = a1, σ(q)(w2)(q)P (w3) + P (w2)σ(q)(w3)(q)(σ0)= a1, P (σ(q)(w2)(q)w3) + P (σ(q)(w3)(q)w2)(σ0) .

The Leibnitz identity for this case now follows from this equality.To prove the sixth case we use the Taylor expansion of P (w2) and P (w3). We

obtain

P (w1), P (w2)P (w3)(σ0)= P (w1), σ(q)(w2)(q)P (w3) + P (w2)σ(q)(w3)(q)(σ0)= P (w1), P (σ(q)(w2)(q)w3) + P (σ(q)(w3)(q)w2)(σ0) .

As before, this equality implies the Leibnitz identity for this case also.

Remark.

In the extreme case V = 0, that is, W ≡ TQ, we have W ∗ = T ∗Q and the Poissonstructure on W ∗ coincides with the standard Poisson structure on the symplecticmanifold T ∗Q. In the other extreme case in which Q is a point, that is W ≡ V ,we have W ∗ = V ∗ and the Poisson structure on W ∗ coincides with the standardLie-Poisson structure on the dual of a Lie algebra.

8.2 Poisson Reduction in the Category LP∗

Viewed as Dual to Reduction in theCategory LP

Let W = TQ⊕ V be an object of LP and assume that the hypotheses of Theorem6.2.10 hold. The composition of the natural vector bundle map πG(W ) : W →W/Gand the vector bundle isomorphism

αWA : (TQ⊕ V )/G→ T (Q/G)⊕ g⊕ (V/G),


of Definition 6.2.11 gives a vector bundle map PG(W ) = αWA πG(W ). Introducingthe notation

WG = T (Q/G)⊕ g⊕ (V/G)

for the target space, WG is an object of LP, and

PG(W ) : W →WG

is a morphism of LP. The restriction of PG(W ) to the zero section coincides withthe natural projection πG(Q) : Q → Q/G and the restriction of PG(W ) to eachfiber is a linear isomorphism. Thus, PG(W ) induces a vector bundle morphismP (W )G : W ∗ → W ∗G. The restriction of P (W )G to the zero section coincides withthe natural projection πG(Q) : Q → Q/G. The restriction of P (W )G to each fiberis a linear isomorphism, which is precisely the dual of the inverse of the restrictionof PG(W ) to the same fiber.

According to Theorem 8.1.3, W ∗ and W ∗G each carry a Poisson structure. We aregoing to show that P (W )G is a Poisson map.

We begin by introducing some notation. Reduced objects under the action of Gwill be denoted with a subindex G. Let us give more precise definitions. For givenw ∈ W we will denote wG = PG(W )(w). Likewise, for µ ∈ W ∗ we will denoteµG = P (W )G(µ). If w ∈ ΓG(W ) or µ ∈ ΓG(W ∗) then, respectively, wG ∈ Γ(WG) isthe unique section of WG such that

PG(W ) (w(q)) = wG (π(q))

for all q ∈ Q and µG ∈ Γ(W ∗G) is the unique section of W ∗G such that

P (W )G (µ(q)) = µG (π(q)) for all q ∈ Q.

If f ∈ C∞(W ∗) is G-invariant or a ∈ C∞(Q) is G-invariant then, respectively, fG ∈C∞(W ∗G) is the unique function such that f = fG P (W )G and aG ∈ C∞(Q/G)is the unique function such that a = aG πG(Q). We can easily show that, forany w ∈ W and any µ ∈ W ∗ we have µG(wG) = µ(w). For given wi ∈ ΓG(W ),i = 1, 2, we have proven before (this is part of the content of Theorem 6.2.10)that [w1G, w2G] = [w1, w2]G. It is also easy to see that for given w ∈ ΓG(W ) anda ∈ C∞(Q), a G-invariant function, we have wG(aG) = (w(a))G. Using the last twoassertions we can easily show that for any given wi ∈ ΓG(W ), i = 1, 2, we have

P (w1G), P (w2G) = P (w1), P (w2)G,

that is, for any σ ∈W ∗ we have

P (w1G), P (w2G)(σG) = P (w1), P (w2)G(σG).

Using the previous notations, the assertion that P (W )G is a Poisson map can berestated as follows: for any given G-invariant functions fi ∈ C∞(W ∗), i = 1, 2, wehave f1G, f2G = f1, f2G.


First, we will consider the case in which fi ∈ A(W ∗), i = 1, 2. Then fi =ai + P (wi), for some ai ∈ C∞(Q) and wi ∈ Γ(W ), i = 1, 2. The G-invarianceof fi, i = 1, 2, implies the G-invariance of ai and of wi, that is, in particular,wi ∈ ΓG(W ), i = 1, 2. Using this and the previous formulas we can show in astraightforward manner that f1G, f2G = f1, f2G.

Second, we will consider the general case. Let fi ∈ C∞(W ∗), i = 1, 2, be G-invariant. Let σ0 ∈ W ∗, say σ0 ∈ W ∗q0 . We can always find σ ∈ ΓG(W ∗) such thatσ(q0) = σ0. We can easily show that fσi ∈ C∞(W ∗) is G-invariant, for i = 1, 2. Letfσi = aσi + P (wσi ), i = 1, 2. We can show easily, for i = 1, 2, that (aσi )G = (aiG)σGand also that (wσi )G = (wiG)σG . Then we have

f1, f2(σ0) = fσ1 , fσ2 (σ0) = fσ1 , fσ2 G(σ0G) = (fσ1 )G, (fσ2 )G(σ0G)= (f1G)σG , (f2G)σG(σ0G) = f1G, f2G(σ0G).

We have proved the following

Theorem 8.2.1. P (W )G : W ∗ →W ∗G is a Poisson map.

This theorem establishes that Poisson reduction in the category LP∗, in par-ticular cotangent bundle reduction, is dual to reduction in the category LP. Thedecomposition of the bracket as a sum of three brackets given in Montgomery,Marsden and Ratiu (1984) and Montgomery (1986) should be compared, via dual-ity, with the decomposition of the reduced Lie bracket on sections of the reducedobject of LP given in Theorem 6.2.10. As we said before, a more detailed study ofall this is being planned for future work.

Acknowledgments.

We thank Darryl Holm, Gerard Misiolek, Matt Perlmutter, Jurgen Scheurle andthe referees for several helpful suggestions.


Page 125

Bibliography

Abraham, R. and J. E. Marsden [1978], Foundations of Mechanics, Addison-Wesley,Second edition.

Abraham, R., J. E. Marsden and T. S. Ratiu [1988], Manifolds, Tensor Analysis andApplications, Applied Mathematical Sciences, 75, Springer-Verlag, New York,Second edition.

Arnold, V. I. [1989], Mathematical Methods of Classical Mechanics, Graduate Textsin Math., 60, Springer-Verlag, Second edition.

Bates, L. and E. Lerman [1997], Proper group actions and symplectic stratifiedspaces, Pacific J. Math., 181, 201–229.

Blaom, A.D. [2000], Reconstruction phases via Poisson reduction, Diff. Geom. andAppl., 12, 231–252.

Bloch, A. M.; Crouch, P. E. [1993] Nonholonomic and vakonomic control systemson Riemannian manifolds. Dynamics and control of mechanical systems, FieldsInst. Commun. 1, 25–52.

Bloch, A. M. and P. E. Crouch [1994], Reduction of Euler–Lagrange problems forconstrained variational problems and relation with optimal control problems, inProc. CDC , 33, 2584–2590.

Bloch, A. M. and P. Crouch [1995], Nonholonomic control systems on Riemannianmanifolds., SIAM J. on Control , 33, 126–148.


Bloch, A. M. and P. E. Crouch [1998], Newton’s law and integrability of nonholo-nomic systems, SIAM J. Control Optim., 6, 2020–2039

Bloch, A. M. and P. E. Crouch [1999], Optimal control, optimization, and an-alytical mechanics, in Mathematical Control Theory , 268–321, J. Ballieul, Ed.,Springer, New York.

Bloch, A. M., P. S. Krishnaprasad, J. E. Marsden and G. Sanchez De Alvarez[1992], Stabilization of rigid body dynamics by internal and external torques,Automatica, 28, 745–756.

Bloch, A. M., P. S. Krishnaprasad, J. E. Marsden and R. Murray [1996], Non-holonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136,21–99.

Bloch, A. M., P. S. Krishnaprasad, J. E. Marsden and T. S. Ratiu [1996], TheEuler–Poincare equations and double bracket dissipation, Comm. Math. Phys.,175, 1–42.

Bloch, A. M., N. Leonard and J. E. Marsden, Controlled Lagrangians and theStabilization of Mechanical Systems I: The First Matching Theorem, IEEE Trans.Automat. Control (to appear).

Bobenko, A. I. and Y. B. Suris [1999a], Discrete time Lagrangian mechanics on Liegroups, with an application to the Lagrange top, Commun. Math. Phys., 204,147–188.

Bobenko, A. I. and Y. B. Suris [1999b], Discrete Lagrangian reduction, discreteEuler-Poincare equations, and semidirect products, Lett. Math. Phys., 49, 79–93.

Bourbaki, N. [1983], Varietes differentielles et analytiqes. Fascicule de resultats,Diffusion C.C.L.S., Paris.

Bretherton, F. P. [1970], A note on Hamilton’s principle for perfect fluids, J. FluidMech., 44, 19–31.

Cannas Da Silva, A. and A. Weinstein [1999], Geometric Models for Noncommuta-tive Alebras, Amer. Math. Soc., Berkeley Mathematics Lecture Notes, 10.

Castrillon Lopez, M., Ratiu, T. S. and Shkoller, S. [2000], Reduction in principalfiber bundles: Covariant Euler-Poincare equations, Proc. Amer. Math. Soc., 128,2155-2164.

Cendra, H., D. D. Holm, M. J. W. Hoyle and J. E. Marsden [1998], The Maxwell–Vlasov equations in Euler–Poincare form, J. Math. Phys., 39, 3138–3157.


Cendra, H., D. D. Holm, J. E. Marsden and T. S. Ratiu [1998], Lagrangian Reduc-tion, the Euler–Poincare Equations and Semidirect Products, Amer. Math. Soc.Transl., 186, 1–25.

Cendra, H., A. Ibort and J. E. Marsden [1987], Variational principal fiber bundles:a geometric theory of Clebsch potentials and Lin constraints, J. Geom. Phys., 4,183–206.

Cendra, H. and J. E. Marsden [1987], Lin constraints, Clebsch potentials and vari-ational principles, Physica D , 27, 63–89.

Cendra, H., J. E. Marsden and T. S. Ratiu [2000], Geometric Mechanics, LagrangianReduction and Nonholonomic Systems, in Mathematics Unlimited , Springer-Verlag, New York.

Courant, T. [1990], Dirac manifolds, Trans. Amer. Math. Soc., 319, 631–661.

De Azcarraga, J.E. and J.M. Izquierdo [1995] Lie Groups, Lie Algebras, Cohomologyand Some Applications in Physics. Cambridge University Press.

Gamboa Saravı, R. E. and J. E. Solomin [2003], On the global version of Euler-Lagrange equations, J. Phys. A 36, 7301–7305.

Ge, Z. and J. E. Marsden [1988], Lie–Poisson integrators and Lie–Poisson Hamilton–Jacobi theory, Phys. Lett. A, 133, 134–139.

Guillemin, V. and S. Sternberg [1980], The moment map and collective motion,Ann. of Phys., 1278, 220–253.

Guillemin, V. and S. Sternberg [1984], Symplectic Techniques in Physics, CambridgeUniversity Press.

Hamel, G. [1904], Die Lagrange–Eulerschen Gleichungen der Mechanik, Z. fur Math-ematik u. Physik , 50, 1–57.

Holm, D. D. [2000], Euler-Poincare Dynamics of Ideal Micropolar Complex Fluids,(preprint).

Holm, D. D., J. E. Marsden and T. S. Ratiu [1986], The Hamiltonian structureof continuum mechanics in material, spatial and convective representations, inSeminaire de Mathematiques superieurs, 100, 11–122, Les Presses de L’Univ. deMontreal.

Holm, D. D., J. E. Marsden and T. S. Ratiu [1998a], The Euler–Poincare equationsand semidirect products with applications to continuum theories, Adv. in Math.,137, 1–8.


Holm, D. D., J. E. Marsden and T. S. Ratiu [1998b], Euler–Poincare models ofideal fluids with nonlinear dispersion, Phys. Rev. Lett., 349, 4173–4177.

Jalnapurkar, S. M., M. Leok, J. E. Marsden and M. West [2000], Discrete Routhreduction; (preprint).

Jalnapurkar, S. M. and J. E. Marsden [1999], Stabilization of Relative EquilibriaII, Regul. Chaotic Dyn., 3, 161–179.

Jalnapurkar, S. M. and J. E. Marsden [2000], Reduction of Hamilton’s VariationalPrinciple, Dynam. Stability Systems, 15, 287–318.

Kane, C, J. E. Marsden, M. Ortiz and M. West [2000], Variational Integrators andthe Newmark Algorithm for Conservative and Dissipative Mechanical Systems,Int. J. Num. Math. Eng., 49, 1295–1325.

Kobayashi, S. and K. Nomizu [1963], Foundations of Differential Geometry, Wiley.

Kolar, I., P.W. Michor, and J. Slovak [1993] Natural Operators in Differential Ge-ometry. Springer-Verlag.

Koon, W. S. and J. E. Marsden [1997a], Optimal control for holonomic and non-holonomic mechanical systems with symmetry and Lagrangian reduction, SIAMJ. Control and Optim., 35, 901–929.

Koon, W. S. and J. E. Marsden [1997b], The geometric structure of nonholonomicmechanics, Proc. CDC , 36, 4856–4862.

Koon, W. S. and J. E. Marsden [1997c], The Hamiltonian and Lagrangian ap-proaches to the dynamics of nonholonomic systems, Rep. Math. Phys., 40, 21–62.

Koon, W. S. and J. E. Marsden [1998], The Poisson reduction of nonholonomicmechanical systems, Reports on Math. Phys., 42, 101–134.

Lagrange, J. L. [1788], Mecanique Analytique, Chez la Veuve Desaint.

Leonard, N. E. and J. E. Marsden [1997], Stability and drift of underwater vehicledynamics: mechanical systems with rigid motion symmetry, Physica D , 105,130–162.

Lewis, A. D. [1996] The geometry of the Gibbs-Appell equations and Gauss’ prin-ciple of least constraint. Rep. Math. Phys. 38, 11–28.

Lewis, A. D. [2000] Towards F = ma in a general setting for Lagrangian mechanics(To appear in Annales Henri Poincare).


Libermann, P. and C. M. Marle [1987], Symplectic Geometry and Analytical Me-chanics, Kluwer Academic Publishers.

Mackenzie, K. [1987] Lie Groupoids and Lie Algebroids in Differntial Geometry .London Math. Soc. Lect. Note Series 124, Cambrigde Univ. Press.

Marsden, J. E. [1992], Lectures on Mechanics, London Math. Soc. Lecture NoteSer., 174, Cambridge University Press.

Marsden, J., G. Misiolek, M. Perlmutter and T. Ratiu [1998], Symplectic reductionfor semidirect products and central extensions, Diff. Geom. and its Appl., 9,173–212.

Marsden, J. E., G. Misiolek, M. Perlmutter and T. S. Ratiu [2000], Reduction bystages and group extensions, Preprint .

Marsden, J. E., R. Montgomery, P. J. Morrison and W. B. Thompson [1986], Co-variant Poisson brackets for classical fields, Annals of Physics, 169, 29–48.

Marsden, J. E., P. J. Morrison and A. Weinstein, The Hamiltonian structure of theBBGKY hierarchy equations, Contemp. Math., AMS , 28, 115–124.

Marsden, J. E., G. W. Patrick and S. Shkoller [1998], Mulltisymplectic Geometry,Variational Integrators and Nonlinear PDEs, Comm. Math. Phys., 199, 351–395.

Marsden, J. E., S. Pekarsky and S. Shkoller [1999], Discrete Euler–Poincare andLie–Poisson equations, Nonlinearity , 12, 1647–1662.

Marsden, J. E., S. Pekarsky and S. Shkoller [2000], Symmetry reduction of discreteLagrangian mechanics on Lie groups, J. Geom. and Phys., 36, 140–151.

Marsden, J. E. and M. Perlmutter [2000], The Orbit Bundle Picture of CotangentBundle Reduction, C. R. Math. Rep. Acad. Sci. Canada, 22, 33–54.

Marsden, J. E. and T. Ratiu [1986], Reduction of Poisson Manifolds, Lett. in Math.Phys., 11, 161–170.

Marsden, J. E. and T. S. Ratiu [1999], Introduction to Mechanics and Symmetry ,Texts in Applied Mathematics, 17; Second Edition, 1999.

Marsden, J. E., T. S. Ratiu and J. Scheurle [2000], Reduction theory and theLagrange-Routh equations, J. Math. Phys., 41, 3379–3429.

Marsden, J. E., T. Ratiu and S. Shkoller [2000], The geometry and analysis of theaveraged Euler equations and a new diffeomorphism group, Geom. Funct. Anal.,10, 582–599.


Marsden, J. E., T. S. Ratiu and A. Weinstein [1984], Semi-direct products andreduction in mechanics, Trans. Amer. Math. Soc., 281, 147–177.

Marsden, J. E., T. S. Ratiu and A. Weinstein [1984], Reduction and Hamiltonianstructures on duals of semidirect product Lie Algebras, Amer. Math. Soc., Prov-idence, RI, Contemp. Math., 28, 55–100.

Marsden, J. E. and J. Scheurle [1993], Lagrangian reduction and the double spher-ical pendulum, ZAMP , 44, 17–43.

Marsden, J. E. and J. Scheurle [1993], The reduced Euler–Lagrange equations,Fields Inst. Commun., 1, 139–164.

Marsden, J. E. and S. Shkoller [1999], Multisymplectic geometry, covariant Hamil-tonians and water waves, Math. Proc. Camb. Phil. Soc., 125, 553–575.

Marsden, J. E. and A. Weinstein [1974], Reduction of symplectic manifolds withsymmetry, Rep. Math. Phys., 5, 121–130.

Meyer, K. R. [1973], Symmetries and integrals in mechanics, in Dynamical Systems,M. Peixoto, ed., 259–273, Academic Press.

Marsden, J. E., R. Montgomery and T. S. Ratiu [1990], Reduction, symmetry andphases in mechanics, Amer. Math. Soc., Providence, RI Memoirs, 436.

Montgomery, R. [1984], Canonical formulations of a particle in a Yang–Mills field,Lett. Math. Phys., 8, 59–67.

Montgomery, R. [1986], The Bundle Picture in Mechanics, Ph.D. Thesis, Universityof California Berkeley.

Montgomery, R. [1990], Isoholonomic problems and some applications, Comm. MathPhys., 128, 565–592.

Montgomery, R. [1993], Gauge theory of the falling cat, Fields Inst. Commun., 1,193–218.

Montgomery, R., J. E. Marsden and T. S. Ratiu [1984], Gauged Lie–Poisson struc-tures, Contemp. Math., 28, 101–114, Amer. Math. Soc., Providence, RI.

Moser, J. and A. P. Veselov [1991], Discrete versions of some classical integrablesystems and factorization of matrix polynomials, Comm. Math. Phys., 139, 217–243.

Ortega, J.-P. [1998], Symmetry, Reduction and Stability in Hamiltonian Systems,Ph.D. Thesis, University of California Santa Cruz.


Ortega, J.-P. and T. S. Ratiu [1997], Persistence and smoothness of critical relativeelements in Hamiltonian systems with symmetry, C. R. Acad. Sci. Paris Ser. IMath., 325, 1107–1111.

Ortega, J.-p. and T. S. Ratiu [2001], Hamiltonian Singular Reduction, Birkhauser,Progress in Math.; (to appear).

Palais, R. S. [1961a], Equivalence of nearby differentiable actions of a compactgroup, Bull. Amer. Math. Soc., 67, 362–364.

Palais, R. S. [1961b], On the existence of slices for actions of non-compact Liegroups, Ann. of Math., 73, 295–323.

Poincare, H. [1901a], Sur une forme nouvelle des equations de la mechanique, C.R. Acad. Sci., 132, 369–371.

Poincare, H. [1901b], Sur la stabilite de l’equilibre des figures piriformes affecteespar une masse fluide en rotation, Philosophical Transactions A, 198, 333–373.

Ratiu, T. S. [1980a], The Euler–Poisson Equations and Integrability, Ph.D. Thesis,University of California at Berkeley.

Ratiu, T. S. [1980b], The motion of the free n-dimensional rigid body, Indiana Univ.Math. Journ., 29, 609–629.

Ratiu, T. S. [1980c], Involution theorems, in Geometric Methods in MathematicalPhysics, G. Kaiser and J. Marsden, eds., Springer Lecture Notes, 775, 219–257.

Ratiu, T. S. [1981], Euler–Poisson equations on Lie algebras and the N -dimensionalheavy rigid body, Proc. Natl. Acad. Sci., USA, 78, 1327–1328.

Ratiu, T. S. [1982a], Euler–Poisson equations on Lie algebras and the N -dimensional heavy rigid body, Amer. J. Math., 104, 409–448, 1337.

Ratiu, T. S. [1982b], The Lie algebraic interpretation of the complete integrabil-ity of the Rosochatius system, in Mathematical Methods in Hydrodynamics andIntegrability in Dynamical Systems (La Jolla Institute, 1981), AIP ConferenceProceedings, 88, 109–116.

Routh, E. J. [1877], Stability of a given state of motion, Halsted Press, New York;Reprinted in Stability of Motion (1975), A. T. Fuller ed.

Sjamaar, R. and E. Lerman [1991], Stratified symplectic spaces and reduction, Ann.of Math., 134, 375–422.


Sternberg, S. [1977], Minimal coupling and the symplectic mechanics of a classicalparticle in the presence of a Yang–Mills field, Proc. Nat. Acad. Sci., 74, 5253–5254.

Tulczyjew, W. M. [1977], The Legendre transformation, Ann. Inst. Poincare, 27,101–114.

Vershik, A. M. and V. Gershkovich [1988], Ya. Nonholonomic problems and thetheory of distributions. Acta Appl. Math. 12, 181–209.

Vershik, A. M. and V. Ya Gershkovich [1994], Non-holonomic Riemannian mani-folds, in Dynamical Systems 7 , Springer, Encyclopaedia of Mathematics, 16.

Veselov, A. P. [1988], Integrable discrete-time systems and difference operators,Funct. An. and Appl., 22, 83–94.

Veselov, A. P. [1991], Integrable Lagrangian correspondences and the factorizationof matrix polynomials, Funct. An. and Appl., 25, 112–123.

Weinstein, A. [1978], A universal phase space for particles in Yang–Mills fields,Lett. Math. Phys., 2, 417–420.

Weinstein, A. [1978], Bifurcations and Hamilton’s principle, Math. Zeit., 159, 235–248.

Weinstein, A. [1983a], The local structure of Poisson manifolds, J. of Diff. Geom.,18, 523–557.

Weinstein, A. [1983b], Sophus Lie and symplectic geometry , Expo. Math., 1, 95–96.

Weinstein, A. [1996], Lagrangian Mechanics and Groupoids, Fields Inst. Commun.,7, 207–231.

Weinstein, A. [1998] Poisson geometry, Diff. Geom. Appl. 9, 213–238.

Wendlandt, J. M. and J. E. Marsden [1997], Mechanical integrators derived from adiscrete variational principle, Physica D , 106, 223–246.

Wong, S. K. [1970], Field and particle equations for the classical Yang–Mills fieldand particles with isotopic spin, Il Nuovo Cimento, LXV, 689–694.

Documents

Lagrangian Reduction by Stages - Caltech Computingmarsden/volume/LRBS/lrbs.pdfPage v Preface This booklet studies the geometry of the reduction of Lagrangian systems with symmetry